L(s) = 1 | − 3.08·3-s − 3.63·5-s − 1.68·7-s + 6.50·9-s − 13-s + 11.2·15-s + 5.53·17-s + 4.83·19-s + 5.18·21-s + 7.77·23-s + 8.22·25-s − 10.8·27-s − 3.11·29-s + 2.63·31-s + 6.11·35-s − 2.81·37-s + 3.08·39-s − 9.39·41-s − 5.33·43-s − 23.6·45-s − 0.477·47-s − 4.16·49-s − 17.0·51-s + 6.15·53-s − 14.9·57-s − 10.6·59-s + 7.06·61-s + ⋯ |
L(s) = 1 | − 1.78·3-s − 1.62·5-s − 0.636·7-s + 2.16·9-s − 0.277·13-s + 2.89·15-s + 1.34·17-s + 1.10·19-s + 1.13·21-s + 1.62·23-s + 1.64·25-s − 2.08·27-s − 0.577·29-s + 0.473·31-s + 1.03·35-s − 0.462·37-s + 0.493·39-s − 1.46·41-s − 0.812·43-s − 3.52·45-s − 0.0696·47-s − 0.595·49-s − 2.39·51-s + 0.845·53-s − 1.97·57-s − 1.38·59-s + 0.904·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4952645543\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4952645543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 + 1.68T + 7T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 - 7.77T + 23T^{2} \) |
| 29 | \( 1 + 3.11T + 29T^{2} \) |
| 31 | \( 1 - 2.63T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 + 5.33T + 43T^{2} \) |
| 47 | \( 1 + 0.477T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 67 | \( 1 + 4.78T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 + 9.53T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + 9.04T + 97T^{2} \) |
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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
−7.73436814590252618738549129657, −7.17065085482841211758056042330, −6.77685313221398424787717149021, −5.83709633323522890448012884053, −5.09650112473040039262390441509, −4.71017057965544334266997241613, −3.59212380168791414094412794098, −3.20237188055184140843196795945, −1.26977913733270547342797704254, −0.46210276071949798820488549152,
0.46210276071949798820488549152, 1.26977913733270547342797704254, 3.20237188055184140843196795945, 3.59212380168791414094412794098, 4.71017057965544334266997241613, 5.09650112473040039262390441509, 5.83709633323522890448012884053, 6.77685313221398424787717149021, 7.17065085482841211758056042330, 7.73436814590252618738549129657