| L(s) = 1 | + 0.214·2-s − 1.95·4-s − 5-s − 3.65·7-s − 0.849·8-s − 0.214·10-s + 0.401·11-s − 13-s − 0.785·14-s + 3.72·16-s − 6.98·17-s + 3.87·19-s + 1.95·20-s + 0.0863·22-s − 5.80·23-s + 25-s − 0.214·26-s + 7.14·28-s + 7.63·29-s − 1.02·31-s + 2.49·32-s − 1.50·34-s + 3.65·35-s − 11.5·37-s + 0.831·38-s + 0.849·40-s + 10.1·41-s + ⋯ |
| L(s) = 1 | + 0.151·2-s − 0.976·4-s − 0.447·5-s − 1.38·7-s − 0.300·8-s − 0.0679·10-s + 0.121·11-s − 0.277·13-s − 0.209·14-s + 0.931·16-s − 1.69·17-s + 0.888·19-s + 0.436·20-s + 0.0183·22-s − 1.20·23-s + 0.200·25-s − 0.0421·26-s + 1.35·28-s + 1.41·29-s − 0.183·31-s + 0.441·32-s − 0.257·34-s + 0.618·35-s − 1.90·37-s + 0.134·38-s + 0.134·40-s + 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7067954454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7067954454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.214T + 2T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 - 0.401T + 11T^{2} \) |
| 17 | \( 1 + 6.98T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 + 5.80T + 23T^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 1.68T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 + 7.59T + 67T^{2} \) |
| 71 | \( 1 + 7.86T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 - 3.29T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.116360658181171498991858543587, −8.812299883540879036445907809772, −7.74140647870427730189130584111, −6.84497726771230921581002042245, −6.11649562395683863177177037447, −5.13819312529780525590215956393, −4.18758845772375589258460049706, −3.57661842991796166427220753233, −2.50231050591668272942149167935, −0.54357029711419994366161390565,
0.54357029711419994366161390565, 2.50231050591668272942149167935, 3.57661842991796166427220753233, 4.18758845772375589258460049706, 5.13819312529780525590215956393, 6.11649562395683863177177037447, 6.84497726771230921581002042245, 7.74140647870427730189130584111, 8.812299883540879036445907809772, 9.116360658181171498991858543587
