| L(s) = 1 | + 6·3-s + 12·5-s + 44·7-s + 27·9-s + 52·11-s − 26·13-s + 72·15-s − 20·17-s − 60·19-s + 264·21-s − 8·23-s + 30·25-s + 108·27-s + 132·29-s − 140·31-s + 312·33-s + 528·35-s + 68·37-s − 156·39-s + 28·41-s + 324·45-s + 36·47-s + 938·49-s − 120·51-s + 668·53-s + 624·55-s − 360·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.07·5-s + 2.37·7-s + 9-s + 1.42·11-s − 0.554·13-s + 1.23·15-s − 0.285·17-s − 0.724·19-s + 2.74·21-s − 0.0725·23-s + 6/25·25-s + 0.769·27-s + 0.845·29-s − 0.811·31-s + 1.64·33-s + 2.54·35-s + 0.302·37-s − 0.640·39-s + 0.106·41-s + 1.07·45-s + 0.111·47-s + 2.73·49-s − 0.329·51-s + 1.73·53-s + 1.52·55-s − 0.836·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.188521139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.188521139\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 44 T + 998 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 20 T + 9238 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 60 T + 10318 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T - 418 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 132 T + 28366 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 140 T + 25782 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 68 T + 68750 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 28 T + 129610 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2914 p T^{2} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 36 T + 201778 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 668 T + 406558 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 508 T + 392026 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 340 T + 471854 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 940 T + 504398 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 300 T - 57006 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1124 T + 1087686 T^{2} - 1124 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1520 T + 1230686 T^{2} + 1520 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 524 T + 908810 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1900 T + 2312266 T^{2} - 1900 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1436 T + 2285142 T^{2} + 1436 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.42214398917128688713621417985, −11.05144205561124725340267116979, −10.31194809942975245233471190816, −10.24788617272732909805315794294, −9.264813226358060540585988578082, −9.215669671319814268434114253472, −8.606010351118545401860918348051, −8.346252110582788825116967335954, −7.54957521691544461779378250866, −7.49378800629612049326858629046, −6.53715315204666169112077639816, −6.23805535267007924477684695878, −5.23710427814114969253752179319, −5.00426301943975155280014032240, −4.15952790984489884125185039394, −3.98523586614477440481232678366, −2.76919953794383634385098601660, −2.10695978885312781704145186954, −1.71775559267031069696382642172, −1.10022464665493037666502970190,
1.10022464665493037666502970190, 1.71775559267031069696382642172, 2.10695978885312781704145186954, 2.76919953794383634385098601660, 3.98523586614477440481232678366, 4.15952790984489884125185039394, 5.00426301943975155280014032240, 5.23710427814114969253752179319, 6.23805535267007924477684695878, 6.53715315204666169112077639816, 7.49378800629612049326858629046, 7.54957521691544461779378250866, 8.346252110582788825116967335954, 8.606010351118545401860918348051, 9.215669671319814268434114253472, 9.264813226358060540585988578082, 10.24788617272732909805315794294, 10.31194809942975245233471190816, 11.05144205561124725340267116979, 11.42214398917128688713621417985
