| L(s) = 1 | − 3-s + 5-s − 3.20·7-s + 9-s − 5.20·11-s + 5.45·13-s − 15-s + 4·17-s − 19-s + 3.20·21-s + 8.65·23-s + 25-s − 27-s − 3.20·29-s + 2·31-s + 5.20·33-s − 3.20·35-s − 11.8·37-s − 5.45·39-s − 9.85·41-s + 3.45·43-s + 45-s − 8.65·47-s + 3.25·49-s − 4·51-s − 10.6·53-s − 5.20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.21·7-s + 0.333·9-s − 1.56·11-s + 1.51·13-s − 0.258·15-s + 0.970·17-s − 0.229·19-s + 0.698·21-s + 1.80·23-s + 0.200·25-s − 0.192·27-s − 0.594·29-s + 0.359·31-s + 0.905·33-s − 0.541·35-s − 1.94·37-s − 0.873·39-s − 1.53·41-s + 0.526·43-s + 0.149·45-s − 1.26·47-s + 0.464·49-s − 0.560·51-s − 1.46·53-s − 0.701·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.237031170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.237031170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 - 8.65T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 9.85T + 41T^{2} \) |
| 43 | \( 1 - 3.45T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 6.90T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 - 8.40T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 0.142T + 97T^{2} \) |
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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
−8.405277707179742552915524993825, −7.48603282466043085512740391034, −6.70805336271148444250210561791, −6.17428763166814308738176458374, −5.37354205278993409376935154125, −4.93729325026524221985319285821, −3.41701869388460455000460819819, −3.21210915509482836846883328993, −1.84703216240304355420527762186, −0.63260094896920489915215132055,
0.63260094896920489915215132055, 1.84703216240304355420527762186, 3.21210915509482836846883328993, 3.41701869388460455000460819819, 4.93729325026524221985319285821, 5.37354205278993409376935154125, 6.17428763166814308738176458374, 6.70805336271148444250210561791, 7.48603282466043085512740391034, 8.405277707179742552915524993825
