| L(s) = 1 | − 8·3-s − 2·5-s − 40·7-s + 32·9-s − 64·11-s − 72·13-s + 16·15-s − 34·17-s + 320·21-s + 96·23-s + 2·25-s − 216·27-s + 170·29-s − 192·31-s + 512·33-s + 80·35-s + 150·37-s + 576·39-s − 190·41-s − 64·45-s − 752·47-s + 800·49-s + 272·51-s + 128·55-s − 914·61-s − 1.28e3·63-s + 144·65-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.178·5-s − 2.15·7-s + 1.18·9-s − 1.75·11-s − 1.53·13-s + 0.275·15-s − 0.485·17-s + 3.32·21-s + 0.870·23-s + 0.0159·25-s − 1.53·27-s + 1.08·29-s − 1.11·31-s + 2.70·33-s + 0.386·35-s + 0.666·37-s + 2.36·39-s − 0.723·41-s − 0.212·45-s − 2.33·47-s + 2.33·49-s + 0.746·51-s + 0.313·55-s − 1.91·61-s − 2.55·63-s + 0.274·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 2 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 40 T + 800 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 64 T + 2048 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13318 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 96 T + 4608 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 170 T + 14450 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 192 T + 18432 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 150 T + 11250 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 190 T + 18050 T^{2} + 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 60010 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 p T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 245770 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 272374 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 914 T + 417698 T^{2} + 914 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 932 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 1264 T + 798848 T^{2} - 1264 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 154 T + 11858 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1016 T + 516128 T^{2} - 1016 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1135110 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1344 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1450 T + 1051250 T^{2} + 1450 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.63022017461656709526458994611, −13.23517080905888502783178857652, −12.81371355649173348600989673351, −12.40131566742496178497061314823, −11.88715825272939641656651430202, −11.18411032541944415621541691005, −10.51023590957071768825396051838, −10.22138480302688206256997304357, −9.570208532429631624592094293510, −9.087121509698385688063070250799, −7.75932687216836605485246630037, −7.38576953583505110183965473877, −6.34618616799128594418456626813, −6.33200513220053938466352161352, −5.09120588829966521474578724692, −4.94732359260043060010763710275, −3.43374754993235699983064716222, −2.58555957893791658117511711681, 0, 0,
2.58555957893791658117511711681, 3.43374754993235699983064716222, 4.94732359260043060010763710275, 5.09120588829966521474578724692, 6.33200513220053938466352161352, 6.34618616799128594418456626813, 7.38576953583505110183965473877, 7.75932687216836605485246630037, 9.087121509698385688063070250799, 9.570208532429631624592094293510, 10.22138480302688206256997304357, 10.51023590957071768825396051838, 11.18411032541944415621541691005, 11.88715825272939641656651430202, 12.40131566742496178497061314823, 12.81371355649173348600989673351, 13.23517080905888502783178857652, 13.63022017461656709526458994611
