| L(s) = 1 | − 1.87·2-s + 3-s + 2.53·4-s − 1.87·6-s − 1.53·7-s − 2.87·8-s + 9-s − 0.347·11-s + 2.53·12-s + 1.87·13-s + 2.87·14-s + 2.87·16-s + 0.347·17-s − 1.87·18-s − 1.53·21-s + 0.652·22-s − 2.87·24-s − 3.53·26-s + 27-s − 3.87·28-s − 29-s − 2.53·32-s − 0.347·33-s − 0.652·34-s + 2.53·36-s + 1.87·39-s + 41-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 3-s + 2.53·4-s − 1.87·6-s − 1.53·7-s − 2.87·8-s + 9-s − 0.347·11-s + 2.53·12-s + 1.87·13-s + 2.87·14-s + 2.87·16-s + 0.347·17-s − 1.87·18-s − 1.53·21-s + 0.652·22-s − 2.87·24-s − 3.53·26-s + 27-s − 3.87·28-s − 29-s − 2.53·32-s − 0.347·33-s − 0.652·34-s + 2.53·36-s + 1.87·39-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6694518067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6694518067\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.87T + T^{2} \) |
| 7 | \( 1 + 1.53T + T^{2} \) |
| 11 | \( 1 + 0.347T + T^{2} \) |
| 13 | \( 1 - 1.87T + T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.53T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.87T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.070886535969787719392027852706, −8.794655490524518078331198545220, −7.86323795441089878409619645631, −7.30625998703915144637154656499, −6.43713359714796277460327852542, −5.88609013077102886402845440676, −3.80980449569617653538362104569, −3.17069456319245308203767496122, −2.23321786640478887660271663919, −1.02541387754140829046272838321,
1.02541387754140829046272838321, 2.23321786640478887660271663919, 3.17069456319245308203767496122, 3.80980449569617653538362104569, 5.88609013077102886402845440676, 6.43713359714796277460327852542, 7.30625998703915144637154656499, 7.86323795441089878409619645631, 8.794655490524518078331198545220, 9.070886535969787719392027852706
