| L(s) = 1 | + 2-s + 0.277·3-s + 4-s + 5-s + 0.277·6-s + 2.55·7-s + 8-s − 2.92·9-s + 10-s − 4.94·11-s + 0.277·12-s + 4.28·13-s + 2.55·14-s + 0.277·15-s + 16-s − 3.28·17-s − 2.92·18-s − 3.94·19-s + 20-s + 0.708·21-s − 4.94·22-s − 6.03·23-s + 0.277·24-s + 25-s + 4.28·26-s − 1.64·27-s + 2.55·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.160·3-s + 0.5·4-s + 0.447·5-s + 0.113·6-s + 0.964·7-s + 0.353·8-s − 0.974·9-s + 0.316·10-s − 1.48·11-s + 0.0801·12-s + 1.18·13-s + 0.682·14-s + 0.0717·15-s + 0.250·16-s − 0.797·17-s − 0.688·18-s − 0.904·19-s + 0.223·20-s + 0.154·21-s − 1.05·22-s − 1.25·23-s + 0.0566·24-s + 0.200·25-s + 0.840·26-s − 0.316·27-s + 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 0.277T + 3T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 23 | \( 1 + 6.03T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 + 3.33T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 - 6.61T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 1.46T + 83T^{2} \) |
| 89 | \( 1 + 7.59T + 89T^{2} \) |
| 97 | \( 1 + 0.655T + 97T^{2} \) |
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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
−7.947263528892518377925914034697, −6.88419172350969598643468348087, −6.07108949316439655659271710877, −5.52571356182780720960619464759, −4.98165139566464138326695345229, −4.04835316357076535555108607253, −3.28013286061321824646870075533, −2.22068903692760228475007785831, −1.83211497999609680517250477100, 0,
1.83211497999609680517250477100, 2.22068903692760228475007785831, 3.28013286061321824646870075533, 4.04835316357076535555108607253, 4.98165139566464138326695345229, 5.52571356182780720960619464759, 6.07108949316439655659271710877, 6.88419172350969598643468348087, 7.947263528892518377925914034697
