| L(s) = 1 | − 176·7-s − 944·13-s + 1.60e4·19-s + 2.92e3·25-s − 2.00e4·31-s − 1.26e5·37-s − 1.80e5·43-s − 2.12e5·49-s − 1.31e5·61-s − 6.35e5·67-s − 8.81e4·73-s + 7.66e5·79-s + 1.66e5·91-s + 3.07e6·97-s + 3.10e6·103-s + 4.59e6·109-s + 2.02e6·121-s + 127-s + 131-s − 2.82e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.513·7-s − 0.429·13-s + 2.34·19-s + 0.187·25-s − 0.674·31-s − 2.49·37-s − 2.26·43-s − 1.80·49-s − 0.581·61-s − 2.11·67-s − 0.226·73-s + 1.55·79-s + 0.220·91-s + 3.37·97-s + 2.84·103-s + 3.55·109-s + 1.14·121-s − 1.20·133-s − 1.86·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.632604726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632604726\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2928 T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 88 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2022354 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 472 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 160544 p^{2} T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8032 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 159706654 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 905902800 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10040 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 63094 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4929189984 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 90064 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14494105410 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26209115280 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 81081215250 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 65946 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 317680 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 111030605730 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 44064 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 383224 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 484333832590 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 986691607872 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1538800 T + p^{6} T^{2} )^{2} \) |
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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
−9.804052260023211105840013674261, −9.757349580343180312487731486670, −9.043961950068416055058782181119, −8.811968640991488111103172701599, −8.254462811304050561255668127539, −7.61195555330067763036612817853, −7.29167385890920115610582189426, −7.04072326220471330165577287257, −6.26235849409854010165473262414, −6.04492766599254263140506985940, −5.20242706221059254902525341696, −5.01993784345217734042099887156, −4.60788894565497682506515396403, −3.50798210552998269246871364151, −3.25999317036573438428316193953, −3.16091674944391922706807259755, −1.87186992737643188220998423982, −1.81614299364587530699169462604, −0.890286902945590878752731099472, −0.30051256009781767259658666445,
0.30051256009781767259658666445, 0.890286902945590878752731099472, 1.81614299364587530699169462604, 1.87186992737643188220998423982, 3.16091674944391922706807259755, 3.25999317036573438428316193953, 3.50798210552998269246871364151, 4.60788894565497682506515396403, 5.01993784345217734042099887156, 5.20242706221059254902525341696, 6.04492766599254263140506985940, 6.26235849409854010165473262414, 7.04072326220471330165577287257, 7.29167385890920115610582189426, 7.61195555330067763036612817853, 8.254462811304050561255668127539, 8.811968640991488111103172701599, 9.043961950068416055058782181119, 9.757349580343180312487731486670, 9.804052260023211105840013674261
