| L(s) = 1 | + 2.97·3-s + 4.30·7-s + 5.87·9-s − 1.96·11-s − 5.83·13-s + 6.46·17-s − 19-s + 12.8·21-s + 4.45·23-s + 8.55·27-s − 5.31·29-s + 0.713·31-s − 5.84·33-s + 8.56·37-s − 17.3·39-s + 2.71·41-s − 7.62·43-s − 4.25·47-s + 11.5·49-s + 19.2·51-s + 7.90·53-s − 2.97·57-s + 6.04·59-s + 1.00·61-s + 25.2·63-s + 12.1·67-s + 13.2·69-s + ⋯ |
| L(s) = 1 | + 1.71·3-s + 1.62·7-s + 1.95·9-s − 0.591·11-s − 1.61·13-s + 1.56·17-s − 0.229·19-s + 2.79·21-s + 0.929·23-s + 1.64·27-s − 0.986·29-s + 0.128·31-s − 1.01·33-s + 1.40·37-s − 2.78·39-s + 0.423·41-s − 1.16·43-s − 0.619·47-s + 1.65·49-s + 2.69·51-s + 1.08·53-s − 0.394·57-s + 0.787·59-s + 0.129·61-s + 3.18·63-s + 1.47·67-s + 1.59·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.947737044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.947737044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 + 5.83T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 + 5.31T + 29T^{2} \) |
| 31 | \( 1 - 0.713T + 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 - 2.71T + 41T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 + 4.25T + 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 59 | \( 1 - 6.04T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 0.272T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 1.95T + 79T^{2} \) |
| 83 | \( 1 + 5.52T + 83T^{2} \) |
| 89 | \( 1 + 0.632T + 89T^{2} \) |
| 97 | \( 1 - 7.10T + 97T^{2} \) |
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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
−7.922815364682524191453376940224, −7.54554243026456583689917716759, −6.92103616307398985347176637734, −5.43753384716445944187799115569, −5.02160445346073841394623540723, −4.25056579922376365654047310300, −3.40244606178760290936239396552, −2.54958345443023443941746005634, −2.07800300941411411628560425522, −1.09002369197568191688607086204,
1.09002369197568191688607086204, 2.07800300941411411628560425522, 2.54958345443023443941746005634, 3.40244606178760290936239396552, 4.25056579922376365654047310300, 5.02160445346073841394623540723, 5.43753384716445944187799115569, 6.92103616307398985347176637734, 7.54554243026456583689917716759, 7.922815364682524191453376940224
