| L(s) = 1 | − 3-s + 1.10·5-s + 5.12·7-s + 9-s + 3.13·11-s + 1.17·13-s − 1.10·15-s − 3.12·17-s + 6.71·19-s − 5.12·21-s + 3.74·23-s − 3.77·25-s − 27-s + 7.57·29-s − 4.19·31-s − 3.13·33-s + 5.68·35-s + 1.50·37-s − 1.17·39-s + 5.64·41-s − 0.425·43-s + 1.10·45-s + 3.62·47-s + 19.2·49-s + 3.12·51-s − 5.71·53-s + 3.47·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.495·5-s + 1.93·7-s + 0.333·9-s + 0.944·11-s + 0.326·13-s − 0.286·15-s − 0.757·17-s + 1.54·19-s − 1.11·21-s + 0.781·23-s − 0.754·25-s − 0.192·27-s + 1.40·29-s − 0.754·31-s − 0.545·33-s + 0.960·35-s + 0.247·37-s − 0.188·39-s + 0.881·41-s − 0.0648·43-s + 0.165·45-s + 0.528·47-s + 2.75·49-s + 0.437·51-s − 0.785·53-s + 0.468·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.845014179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.845014179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 - 1.50T + 37T^{2} \) |
| 41 | \( 1 - 5.64T + 41T^{2} \) |
| 43 | \( 1 + 0.425T + 43T^{2} \) |
| 47 | \( 1 - 3.62T + 47T^{2} \) |
| 53 | \( 1 + 5.71T + 53T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 + 2.49T + 61T^{2} \) |
| 67 | \( 1 + 6.96T + 67T^{2} \) |
| 71 | \( 1 - 4.75T + 71T^{2} \) |
| 73 | \( 1 + 7.61T + 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 + 6.48T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.986544954977170220167431644341, −7.39673254018275320812843028389, −6.64193791010978361229601329150, −5.81583986586578031646441579705, −5.21343340224385355852627514423, −4.58235717039834544522439871601, −3.89203628761962636962159380328, −2.60498667511174515378531835774, −1.56356010373038249696066857310, −1.06389475257740601832372584145,
1.06389475257740601832372584145, 1.56356010373038249696066857310, 2.60498667511174515378531835774, 3.89203628761962636962159380328, 4.58235717039834544522439871601, 5.21343340224385355852627514423, 5.81583986586578031646441579705, 6.64193791010978361229601329150, 7.39673254018275320812843028389, 7.986544954977170220167431644341
