| L(s) = 1 | − 0.788·2-s + 1.45·3-s − 1.37·4-s − 3.12·5-s − 1.14·6-s + 4.52·7-s + 2.66·8-s − 0.888·9-s + 2.46·10-s − 3.73·11-s − 2.00·12-s − 1.95·13-s − 3.56·14-s − 4.53·15-s + 0.659·16-s − 3.59·17-s + 0.700·18-s − 3.52·19-s + 4.30·20-s + 6.58·21-s + 2.94·22-s + 6.52·23-s + 3.86·24-s + 4.75·25-s + 1.54·26-s − 5.65·27-s − 6.24·28-s + ⋯ |
| L(s) = 1 | − 0.557·2-s + 0.838·3-s − 0.689·4-s − 1.39·5-s − 0.467·6-s + 1.71·7-s + 0.941·8-s − 0.296·9-s + 0.778·10-s − 1.12·11-s − 0.578·12-s − 0.543·13-s − 0.953·14-s − 1.17·15-s + 0.164·16-s − 0.872·17-s + 0.165·18-s − 0.809·19-s + 0.963·20-s + 1.43·21-s + 0.627·22-s + 1.36·23-s + 0.789·24-s + 0.951·25-s + 0.302·26-s − 1.08·27-s − 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 619 | \( 1 + T \) |
| good | 2 | \( 1 + 0.788T + 2T^{2} \) |
| 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 + 3.12T + 5T^{2} \) |
| 7 | \( 1 - 4.52T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 + 1.95T + 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 9.35T + 41T^{2} \) |
| 43 | \( 1 + 9.90T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2.07T + 53T^{2} \) |
| 59 | \( 1 - 9.34T + 59T^{2} \) |
| 61 | \( 1 + 3.09T + 61T^{2} \) |
| 67 | \( 1 + 7.45T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 - 9.81T + 97T^{2} \) |
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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
−10.13319217487528815744272379060, −8.860988630302386471098584043511, −8.397124569366898568299024461402, −7.902588596208953785165901907222, −7.25397600271348738563426844654, −5.09403311959361271903229350477, −4.63026060325206169383233656782, −3.46870515975175331441157665901, −2.00402742848736068217502159225, 0,
2.00402742848736068217502159225, 3.46870515975175331441157665901, 4.63026060325206169383233656782, 5.09403311959361271903229350477, 7.25397600271348738563426844654, 7.902588596208953785165901907222, 8.397124569366898568299024461402, 8.860988630302386471098584043511, 10.13319217487528815744272379060
