| L(s) = 1 | − 3-s − 3.34·5-s + 2.12·7-s + 9-s − 5.44·11-s + 3.34·15-s − 6.68·17-s − 7.70·19-s − 2.12·21-s + 7.16·23-s + 6.20·25-s − 27-s − 2.79·29-s + 1.30·31-s + 5.44·33-s − 7.12·35-s + 6.38·37-s − 9.98·41-s − 1.95·43-s − 3.34·45-s − 5.32·47-s − 2.46·49-s + 6.68·51-s + 11.2·53-s + 18.2·55-s + 7.70·57-s + 6.53·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.49·5-s + 0.804·7-s + 0.333·9-s − 1.64·11-s + 0.864·15-s − 1.62·17-s − 1.76·19-s − 0.464·21-s + 1.49·23-s + 1.24·25-s − 0.192·27-s − 0.518·29-s + 0.234·31-s + 0.947·33-s − 1.20·35-s + 1.04·37-s − 1.55·41-s − 0.298·43-s − 0.498·45-s − 0.776·47-s − 0.352·49-s + 0.935·51-s + 1.54·53-s + 2.45·55-s + 1.02·57-s + 0.850·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4615932399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4615932399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 + 5.44T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 + 7.70T + 19T^{2} \) |
| 23 | \( 1 - 7.16T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 - 1.30T + 31T^{2} \) |
| 37 | \( 1 - 6.38T + 37T^{2} \) |
| 41 | \( 1 + 9.98T + 41T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 + 5.32T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 6.53T + 59T^{2} \) |
| 61 | \( 1 + 7.80T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 + 3.81T + 71T^{2} \) |
| 73 | \( 1 - 3.67T + 73T^{2} \) |
| 79 | \( 1 + 3.08T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.507077302414488836238645070481, −7.67484246511948822228824065258, −7.10018413942813103369190648997, −6.34074080688541677051518631792, −5.18665412392188899636937916826, −4.66948606360698463112137551732, −4.12818089523744210815080431629, −2.95846104250877759200457148009, −1.97747572859788986895662137123, −0.38052015499644186472308026327,
0.38052015499644186472308026327, 1.97747572859788986895662137123, 2.95846104250877759200457148009, 4.12818089523744210815080431629, 4.66948606360698463112137551732, 5.18665412392188899636937916826, 6.34074080688541677051518631792, 7.10018413942813103369190648997, 7.67484246511948822228824065258, 8.507077302414488836238645070481
