| L(s) = 1 | − 2-s + 4-s − 2.82·5-s − 8-s − 3·9-s + 2.82·10-s + 11-s − 4.24·13-s + 16-s + 2.82·17-s + 3·18-s − 1.41·19-s − 2.82·20-s − 22-s + 6·23-s + 3.00·25-s + 4.24·26-s + 8·29-s + 1.41·31-s − 32-s − 2.82·34-s − 3·36-s − 6·37-s + 1.41·38-s + 2.82·40-s − 8.48·41-s + 10·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.26·5-s − 0.353·8-s − 9-s + 0.894·10-s + 0.301·11-s − 1.17·13-s + 0.250·16-s + 0.685·17-s + 0.707·18-s − 0.324·19-s − 0.632·20-s − 0.213·22-s + 1.25·23-s + 0.600·25-s + 0.832·26-s + 1.48·29-s + 0.254·31-s − 0.176·32-s − 0.485·34-s − 0.5·36-s − 0.986·37-s + 0.229·38-s + 0.447·40-s − 1.32·41-s + 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6761349076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6761349076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 - 1.41T + 97T^{2} \) |
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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
−9.898290459616744599754825348698, −8.774048791834705073474445509367, −8.411640667493256185691590438354, −7.42041539660764684964341793639, −6.90598034759782760499441213991, −5.64558750702522696840066099678, −4.64171572849634868423435587742, −3.44685398696926283075448117350, −2.54473546694734854155889576987, −0.67651464646677469676777367322,
0.67651464646677469676777367322, 2.54473546694734854155889576987, 3.44685398696926283075448117350, 4.64171572849634868423435587742, 5.64558750702522696840066099678, 6.90598034759782760499441213991, 7.42041539660764684964341793639, 8.411640667493256185691590438354, 8.774048791834705073474445509367, 9.898290459616744599754825348698
