| L(s) = 1 | + 3.14·3-s + 0.677·5-s + 1.67·7-s + 6.91·9-s + 11-s − 4.16·13-s + 2.13·15-s + 6.86·17-s − 19-s + 5.27·21-s − 2.14·23-s − 4.54·25-s + 12.3·27-s + 2.01·29-s + 2.13·31-s + 3.14·33-s + 1.13·35-s + 2.81·37-s − 13.1·39-s + 10.9·41-s − 3.39·43-s + 4.67·45-s − 1.22·47-s − 4.18·49-s + 21.6·51-s − 11.7·53-s + 0.677·55-s + ⋯ |
| L(s) = 1 | + 1.81·3-s + 0.302·5-s + 0.633·7-s + 2.30·9-s + 0.301·11-s − 1.15·13-s + 0.550·15-s + 1.66·17-s − 0.229·19-s + 1.15·21-s − 0.446·23-s − 0.908·25-s + 2.36·27-s + 0.373·29-s + 0.382·31-s + 0.548·33-s + 0.191·35-s + 0.462·37-s − 2.09·39-s + 1.71·41-s − 0.517·43-s + 0.697·45-s − 0.179·47-s − 0.598·49-s + 3.02·51-s − 1.61·53-s + 0.0913·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.415785714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.415785714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 5 | \( 1 - 0.677T + 5T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 17 | \( 1 - 6.86T + 17T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 - 2.13T + 31T^{2} \) |
| 37 | \( 1 - 2.81T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 3.39T + 43T^{2} \) |
| 47 | \( 1 + 1.22T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 8.37T + 59T^{2} \) |
| 61 | \( 1 - 9.52T + 61T^{2} \) |
| 67 | \( 1 - 9.64T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 + 4.08T + 89T^{2} \) |
| 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
−8.490095165689655548522333772429, −7.84614526523453469426410392711, −7.60179746568786621160836973257, −6.58414588295287906453531273533, −5.49086350371220698396411532095, −4.55884610933526606199770485095, −3.81363550021013917763536756538, −2.91096512659384949506533354962, −2.19637289188845690605459435022, −1.30701342155216906148581503521,
1.30701342155216906148581503521, 2.19637289188845690605459435022, 2.91096512659384949506533354962, 3.81363550021013917763536756538, 4.55884610933526606199770485095, 5.49086350371220698396411532095, 6.58414588295287906453531273533, 7.60179746568786621160836973257, 7.84614526523453469426410392711, 8.490095165689655548522333772429
