| L(s) = 1 | + 29·4-s − 42·5-s + 136·7-s + 23·9-s + 222·11-s + 283·16-s + 300·17-s + 114·19-s − 1.21e3·20-s − 78·23-s − 1.36e3·25-s + 3.94e3·28-s − 5.71e3·35-s + 667·36-s + 2.98e3·43-s + 6.43e3·44-s − 966·45-s − 7.57e3·47-s + 3.27e3·49-s − 9.32e3·55-s + 158·61-s + 3.12e3·63-s − 551·64-s + 8.70e3·68-s − 1.51e4·73-s + 3.30e3·76-s + 3.01e4·77-s + ⋯ |
| L(s) = 1 | + 1.81·4-s − 1.67·5-s + 2.77·7-s + 0.283·9-s + 1.83·11-s + 1.10·16-s + 1.03·17-s + 6/19·19-s − 3.04·20-s − 0.147·23-s − 2.17·25-s + 5.03·28-s − 4.66·35-s + 0.514·36-s + 1.61·43-s + 3.32·44-s − 0.477·45-s − 3.43·47-s + 1.36·49-s − 3.08·55-s + 0.0424·61-s + 0.788·63-s − 0.134·64-s + 1.88·68-s − 2.84·73-s + 0.572·76-s + 5.09·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130321 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130321 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.003238657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.003238657\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 19 | $D_{4}$ | \( 1 - 6 p T + 74 p^{2} T^{2} - 6 p^{5} T^{3} + p^{8} T^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - 29 T^{2} + 279 p T^{4} - 29 p^{8} T^{6} + p^{16} T^{8} \) |
| 3 | $C_2^2 \wr C_2$ | \( 1 - 23 T^{2} - 476 p^{2} T^{4} - 23 p^{8} T^{6} + p^{16} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 21 T + 1342 T^{2} + 21 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 - 68 T + 5301 T^{2} - 68 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 111 T + 27088 T^{2} - 111 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 77183 T^{2} + 3054117756 T^{4} - 77183 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 150 T + 162155 T^{2} - 150 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 39 T + 327028 T^{2} + 39 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1295951 T^{2} + 1154150796468 T^{4} - 1295951 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 348056 T^{2} + 1510187432718 T^{4} - 348056 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 3680824 T^{2} + 8018883544974 T^{4} - 3680824 p^{8} T^{6} + p^{16} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 4843928 T^{2} + 12077160453006 T^{4} - 4843928 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1493 T + 5146446 T^{2} - 1493 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 3789 T + 13216636 T^{2} + 3789 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 28576543 T^{2} + 328095076878996 T^{4} - 28576543 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 20616935 T^{2} + 353317496460300 T^{4} - 20616935 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 79 T + 27159594 T^{2} - 79 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 69010271 T^{2} + 1992542074732308 T^{4} - 69010271 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 41313424 T^{2} + 12545129831538 p T^{4} - 41313424 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 7584 T + 69041153 T^{2} + 7584 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 1822376 p T^{2} + 8180923573778574 T^{4} - 1822376 p^{9} T^{6} + p^{16} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 16638 T + 162869650 T^{2} - 16638 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 52179052 T^{2} + 8356348223824470 T^{4} - 52179052 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 337141424 T^{2} + 44087261433798366 T^{4} - 337141424 p^{8} T^{6} + p^{16} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.32520667665192961893481826871, −12.80171011224918103454268168571, −11.93763691418807760078785686073, −11.93602883722589436693747057571, −11.85363818440540829175892094423, −11.53054489930532385488169574777, −11.36544530996201354536132130058, −10.97974936158881818628481306518, −10.64607847711212504776886181858, −9.985120073947558333567265117206, −9.468481148024772974935029083733, −9.134075433092274029135094880537, −8.114077941291070242756339539577, −8.043768307758072156766810939729, −7.889404077562547067488969293782, −7.53114571878574115368892188680, −6.95844539013092255267451682695, −6.42450242004923506748029341893, −5.96778001144018337658640938946, −5.13394310412386057780785896548, −4.52661543118466984280265281182, −3.98684889013541380180422930275, −3.40893990867664528676093416735, −1.84738697150198144510601938716, −1.58256861612102069882584949014,
1.58256861612102069882584949014, 1.84738697150198144510601938716, 3.40893990867664528676093416735, 3.98684889013541380180422930275, 4.52661543118466984280265281182, 5.13394310412386057780785896548, 5.96778001144018337658640938946, 6.42450242004923506748029341893, 6.95844539013092255267451682695, 7.53114571878574115368892188680, 7.889404077562547067488969293782, 8.043768307758072156766810939729, 8.114077941291070242756339539577, 9.134075433092274029135094880537, 9.468481148024772974935029083733, 9.985120073947558333567265117206, 10.64607847711212504776886181858, 10.97974936158881818628481306518, 11.36544530996201354536132130058, 11.53054489930532385488169574777, 11.85363818440540829175892094423, 11.93602883722589436693747057571, 11.93763691418807760078785686073, 12.80171011224918103454268168571, 13.32520667665192961893481826871
