| L(s) = 1 | + 5·5-s + 31.3·7-s − 8.94·11-s − 62·13-s + 46·17-s − 107.·19-s + 192.·23-s + 25·25-s + 90·29-s + 152.·31-s + 156.·35-s − 214·37-s + 10·41-s + 67.0·43-s + 398.·47-s + 637.·49-s + 678·53-s − 44.7·55-s − 411.·59-s + 250·61-s − 310·65-s − 49.1·67-s − 366.·71-s + 522·73-s − 280·77-s − 876.·79-s + 380.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.69·7-s − 0.245·11-s − 1.32·13-s + 0.656·17-s − 1.29·19-s + 1.74·23-s + 0.200·25-s + 0.576·29-s + 0.880·31-s + 0.755·35-s − 0.950·37-s + 0.0380·41-s + 0.237·43-s + 1.23·47-s + 1.85·49-s + 1.75·53-s − 0.109·55-s − 0.907·59-s + 0.524·61-s − 0.591·65-s − 0.0897·67-s − 0.612·71-s + 0.836·73-s − 0.414·77-s − 1.24·79-s + 0.502·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.901097428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.901097428\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 + 8.94T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214T + 5.06e4T^{2} \) |
| 41 | \( 1 - 10T + 6.89e4T^{2} \) |
| 43 | \( 1 - 67.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 678T + 1.48e5T^{2} \) |
| 59 | \( 1 + 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250T + 2.26e5T^{2} \) |
| 67 | \( 1 + 49.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522T + 3.89e5T^{2} \) |
| 79 | \( 1 + 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 970T + 7.04e5T^{2} \) |
| 97 | \( 1 + 934T + 9.12e5T^{2} \) |
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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
−8.983627928779647567816298626939, −8.427838331288779887210326280015, −7.54135544425348171218500778432, −6.89884173275173622235119617998, −5.64104990070064938606814840571, −4.96114164849100291571240614971, −4.35559868509634142516369702348, −2.79402996792555998509872273302, −1.97741141288338885018042808299, −0.867507139521931367981835595353,
0.867507139521931367981835595353, 1.97741141288338885018042808299, 2.79402996792555998509872273302, 4.35559868509634142516369702348, 4.96114164849100291571240614971, 5.64104990070064938606814840571, 6.89884173275173622235119617998, 7.54135544425348171218500778432, 8.427838331288779887210326280015, 8.983627928779647567816298626939
