| L(s) = 1 | + 0.615·3-s + 0.179·5-s + 1.85·7-s − 2.62·9-s + 11-s + 0.370·13-s + 0.110·15-s − 1.41·17-s + 3.88·19-s + 1.14·21-s − 4.56·23-s − 4.96·25-s − 3.45·27-s − 8.96·29-s + 4.49·31-s + 0.615·33-s + 0.333·35-s − 0.121·37-s + 0.227·39-s − 11.6·41-s + 6.05·43-s − 0.471·45-s + 5.76·47-s − 3.55·49-s − 0.873·51-s − 10.1·53-s + 0.179·55-s + ⋯ |
| L(s) = 1 | + 0.355·3-s + 0.0804·5-s + 0.701·7-s − 0.873·9-s + 0.301·11-s + 0.102·13-s + 0.0285·15-s − 0.344·17-s + 0.891·19-s + 0.249·21-s − 0.951·23-s − 0.993·25-s − 0.665·27-s − 1.66·29-s + 0.807·31-s + 0.107·33-s + 0.0564·35-s − 0.0200·37-s + 0.0364·39-s − 1.81·41-s + 0.924·43-s − 0.0702·45-s + 0.840·47-s − 0.507·49-s − 0.122·51-s − 1.39·53-s + 0.0242·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 0.615T + 3T^{2} \) |
| 5 | \( 1 - 0.179T + 5T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 13 | \( 1 - 0.370T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 3.88T + 19T^{2} \) |
| 23 | \( 1 + 4.56T + 23T^{2} \) |
| 29 | \( 1 + 8.96T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 + 0.121T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 6.05T + 43T^{2} \) |
| 47 | \( 1 - 5.76T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 0.171T + 59T^{2} \) |
| 61 | \( 1 + 3.59T + 61T^{2} \) |
| 67 | \( 1 + 1.63T + 67T^{2} \) |
| 71 | \( 1 + 0.291T + 71T^{2} \) |
| 73 | \( 1 - 2.58T + 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7.94T + 97T^{2} \) |
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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
−7.897130281234723097523734398514, −7.16228730116701100867913820571, −6.15543919104668851076764729501, −5.64746545746476570320391894129, −4.85594146012006253343586498755, −3.95347707054783493330306661379, −3.25913159681944534282797880237, −2.25836699117237495205652543731, −1.51075811958411661016278942186, 0,
1.51075811958411661016278942186, 2.25836699117237495205652543731, 3.25913159681944534282797880237, 3.95347707054783493330306661379, 4.85594146012006253343586498755, 5.64746545746476570320391894129, 6.15543919104668851076764729501, 7.16228730116701100867913820571, 7.897130281234723097523734398514
