| L(s) = 1 | + 326·19-s − 578·31-s − 683·49-s + 1.43e3·61-s − 1.76e3·79-s + 3.13e3·109-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.52e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 3.93·19-s − 3.34·31-s − 1.99·49-s + 3.01·61-s − 2.51·79-s + 2.75·109-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.60·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.418898437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.418898437\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 683 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3527 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 163 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 289 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 89206 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 153973 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 719 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 412523 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 638066 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1551817 T^{2} + p^{6} T^{4} \) |
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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
−9.697442349553455291774931397408, −9.661108318046026488170777793855, −9.248725018576262941382477108833, −8.661526444259823005630614260594, −8.331837986778780954958700938764, −7.52986692699608835136167042689, −7.44280408206148735348245766181, −7.19542341824109355706836894401, −6.59598585055806275264960064161, −5.77401986209207217738621203079, −5.64232201032660971204691390019, −5.02082887477455740315305275488, −4.95502612195815608294110644936, −3.75438714102665352192576391522, −3.67882723062805695100255053576, −3.12258795519113445520578147710, −2.51771987921337726678921299958, −1.62851991666910095991270693425, −1.25105663711596457826280928176, −0.42590873561650351997270247669,
0.42590873561650351997270247669, 1.25105663711596457826280928176, 1.62851991666910095991270693425, 2.51771987921337726678921299958, 3.12258795519113445520578147710, 3.67882723062805695100255053576, 3.75438714102665352192576391522, 4.95502612195815608294110644936, 5.02082887477455740315305275488, 5.64232201032660971204691390019, 5.77401986209207217738621203079, 6.59598585055806275264960064161, 7.19542341824109355706836894401, 7.44280408206148735348245766181, 7.52986692699608835136167042689, 8.331837986778780954958700938764, 8.661526444259823005630614260594, 9.248725018576262941382477108833, 9.661108318046026488170777793855, 9.697442349553455291774931397408
