| L(s) = 1 | + 2.90·3-s − 5-s − 1.52·7-s + 5.42·9-s − 4.90·11-s − 6.42·13-s − 2.90·15-s + 2.14·17-s − 2.28·19-s − 4.42·21-s − 6.90·23-s + 25-s + 7.05·27-s + 29-s − 1.71·31-s − 14.2·33-s + 1.52·35-s + 7.95·37-s − 18.6·39-s − 3.37·41-s + 1.09·43-s − 5.42·45-s − 12.7·47-s − 4.67·49-s + 6.23·51-s + 3.37·53-s + 4.90·55-s + ⋯ |
| L(s) = 1 | + 1.67·3-s − 0.447·5-s − 0.576·7-s + 1.80·9-s − 1.47·11-s − 1.78·13-s − 0.749·15-s + 0.520·17-s − 0.523·19-s − 0.966·21-s − 1.43·23-s + 0.200·25-s + 1.35·27-s + 0.185·29-s − 0.308·31-s − 2.47·33-s + 0.257·35-s + 1.30·37-s − 2.98·39-s − 0.527·41-s + 0.167·43-s − 0.809·45-s − 1.85·47-s − 0.667·49-s + 0.873·51-s + 0.463·53-s + 0.661·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 + 6.42T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 - 7.95T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 3.37T + 53T^{2} \) |
| 59 | \( 1 - 3.18T + 59T^{2} \) |
| 61 | \( 1 + 2.42T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 + 3.57T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 0.341T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−8.387990829881667670066576221086, −7.88002495231077656060814451814, −7.50038557886195642778481924766, −6.52055702165479018622212677102, −5.24853358501967100201659016714, −4.41116613196407847411868033492, −3.45396964382149648558793537038, −2.70050274549717620539678227744, −2.09601458715400040731379040194, 0,
2.09601458715400040731379040194, 2.70050274549717620539678227744, 3.45396964382149648558793537038, 4.41116613196407847411868033492, 5.24853358501967100201659016714, 6.52055702165479018622212677102, 7.50038557886195642778481924766, 7.88002495231077656060814451814, 8.387990829881667670066576221086
