| L(s) = 1 | − 2-s + 4-s + 3·5-s + 7-s − 8-s − 3·10-s + 5·13-s − 14-s + 16-s + 8·19-s + 3·20-s + 23-s + 4·25-s − 5·26-s + 28-s − 3·29-s + 2·31-s − 32-s + 3·35-s − 7·37-s − 8·38-s − 3·40-s − 9·41-s − 43-s − 46-s + 3·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s − 0.353·8-s − 0.948·10-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.83·19-s + 0.670·20-s + 0.208·23-s + 4/5·25-s − 0.980·26-s + 0.188·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s + 0.507·35-s − 1.15·37-s − 1.29·38-s − 0.474·40-s − 1.40·41-s − 0.152·43-s − 0.147·46-s + 0.437·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084924735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084924735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.821317656689202549443528913758, −8.232524183544560860688464311829, −7.21168097014111131891212362730, −6.60811491963455317329179529961, −5.56133283177033100824366654679, −5.38147887888854304412534738693, −3.85236343837231962091875263875, −2.89077933654956654393036807338, −1.78994296405010944893448177210, −1.09857844246470733126369777529,
1.09857844246470733126369777529, 1.78994296405010944893448177210, 2.89077933654956654393036807338, 3.85236343837231962091875263875, 5.38147887888854304412534738693, 5.56133283177033100824366654679, 6.60811491963455317329179529961, 7.21168097014111131891212362730, 8.232524183544560860688464311829, 8.821317656689202549443528913758
