| L(s) = 1 | + 12·4-s + 29·9-s + 68·11-s + 80·16-s + 140·19-s − 490·29-s + 206·31-s + 348·36-s + 190·41-s + 816·44-s + 622·49-s + 816·59-s + 1.64e3·61-s + 192·64-s + 670·71-s + 1.68e3·76-s + 2.64e3·79-s + 112·81-s + 920·89-s + 1.97e3·99-s − 620·101-s − 1.40e3·109-s − 5.88e3·116-s + 806·121-s + 2.47e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1.07·9-s + 1.86·11-s + 5/4·16-s + 1.69·19-s − 3.13·29-s + 1.19·31-s + 1.61·36-s + 0.723·41-s + 2.79·44-s + 1.81·49-s + 1.80·59-s + 3.45·61-s + 3/8·64-s + 1.11·71-s + 2.53·76-s + 3.76·79-s + 0.153·81-s + 1.09·89-s + 2.00·99-s − 0.610·101-s − 1.23·109-s − 4.70·116-s + 0.605·121-s + 1.79·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.480819286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.480819286\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p^{2} T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1145 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3426 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 245 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 103 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 12502 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 95 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 151270 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 80197 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 408 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 822 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 255950 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 335 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 30167 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1322 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1142278 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 460 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 896050 T^{2} + p^{6} T^{4} \) |
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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
−10.43749964819396897792816458512, −10.23132894454394357122530824027, −9.558971416604783464676530835586, −9.338941710641117380520438016536, −9.030269423578718539116638444389, −8.118822081862201329210320790935, −7.74914084702065552915789797540, −7.28356960470888393957735300881, −6.87358505207142187880071137572, −6.70696221129548232724452858336, −6.12116551003806861301770201128, −5.45411287110198440693907782189, −5.21855581204011814752698866323, −4.14460137014459061958063632543, −3.71157202243366749369338326063, −3.55016546655196146135037886220, −2.28232646954849400118117461266, −2.20327613247934851481571818358, −1.14384025185583879611576778850, −0.989664047375344293887385998698,
0.989664047375344293887385998698, 1.14384025185583879611576778850, 2.20327613247934851481571818358, 2.28232646954849400118117461266, 3.55016546655196146135037886220, 3.71157202243366749369338326063, 4.14460137014459061958063632543, 5.21855581204011814752698866323, 5.45411287110198440693907782189, 6.12116551003806861301770201128, 6.70696221129548232724452858336, 6.87358505207142187880071137572, 7.28356960470888393957735300881, 7.74914084702065552915789797540, 8.118822081862201329210320790935, 9.030269423578718539116638444389, 9.338941710641117380520438016536, 9.558971416604783464676530835586, 10.23132894454394357122530824027, 10.43749964819396897792816458512
