| L(s) = 1 | + 2·3-s + 4·5-s + 3·9-s − 4·11-s + 4·13-s + 8·15-s + 4·17-s + 8·23-s + 4·25-s + 4·27-s + 12·29-s − 8·33-s + 12·37-s + 8·39-s − 4·41-s + 8·43-s + 12·45-s + 8·47-s − 12·49-s + 8·51-s + 12·53-s − 16·55-s − 8·59-s + 12·61-s + 16·65-s + 16·69-s + 8·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 9-s − 1.20·11-s + 1.10·13-s + 2.06·15-s + 0.970·17-s + 1.66·23-s + 4/5·25-s + 0.769·27-s + 2.22·29-s − 1.39·33-s + 1.97·37-s + 1.28·39-s − 0.624·41-s + 1.21·43-s + 1.78·45-s + 1.16·47-s − 1.71·49-s + 1.12·51-s + 1.64·53-s − 2.15·55-s − 1.04·59-s + 1.53·61-s + 1.98·65-s + 1.92·69-s + 0.923·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.542148911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.542148911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 32 T + 412 T^{2} + 32 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.694468455169911339648426407487, −9.263247525888969910434106896920, −8.726242909827584481788428541958, −8.680088304946863075078983045293, −8.034177794536019404695437551618, −7.902143102458363450844213594938, −7.08914896736604951814659432982, −7.06217321280785590516066329506, −6.26083503226687910235809933465, −6.03927125108928665263293464136, −5.48865383006853673783369906253, −5.32388382876011597818427962489, −4.52854011525196972213481184143, −4.27645982070683424710230626549, −3.47215109372313747215187451736, −2.88428709462737857148108864879, −2.67676023609515521262976787307, −2.28089429096443038666811498393, −1.22361178559583339778249123595, −1.20272965852503917689697692874,
1.20272965852503917689697692874, 1.22361178559583339778249123595, 2.28089429096443038666811498393, 2.67676023609515521262976787307, 2.88428709462737857148108864879, 3.47215109372313747215187451736, 4.27645982070683424710230626549, 4.52854011525196972213481184143, 5.32388382876011597818427962489, 5.48865383006853673783369906253, 6.03927125108928665263293464136, 6.26083503226687910235809933465, 7.06217321280785590516066329506, 7.08914896736604951814659432982, 7.902143102458363450844213594938, 8.034177794536019404695437551618, 8.680088304946863075078983045293, 8.726242909827584481788428541958, 9.263247525888969910434106896920, 9.694468455169911339648426407487
