| L(s) = 1 | − 1.67·3-s + 5-s + 4.15·7-s − 0.193·9-s − 0.806·11-s + 0.481·13-s − 1.67·15-s − 3.61·17-s − 19-s − 6.96·21-s − 5.76·23-s + 25-s + 5.35·27-s − 8.57·29-s + 3.61·31-s + 1.35·33-s + 4.15·35-s − 9.18·37-s − 0.806·39-s + 0.312·41-s − 3.19·43-s − 0.193·45-s − 5.76·47-s + 10.2·49-s + 6.05·51-s − 8.21·53-s − 0.806·55-s + ⋯ |
| L(s) = 1 | − 0.967·3-s + 0.447·5-s + 1.57·7-s − 0.0646·9-s − 0.243·11-s + 0.133·13-s − 0.432·15-s − 0.876·17-s − 0.229·19-s − 1.51·21-s − 1.20·23-s + 0.200·25-s + 1.02·27-s − 1.59·29-s + 0.648·31-s + 0.235·33-s + 0.702·35-s − 1.50·37-s − 0.129·39-s + 0.0488·41-s − 0.487·43-s − 0.0289·45-s − 0.841·47-s + 1.46·49-s + 0.847·51-s − 1.12·53-s − 0.108·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 11 | \( 1 + 0.806T + 11T^{2} \) |
| 13 | \( 1 - 0.481T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 - 3.61T + 31T^{2} \) |
| 37 | \( 1 + 9.18T + 37T^{2} \) |
| 41 | \( 1 - 0.312T + 41T^{2} \) |
| 43 | \( 1 + 3.19T + 43T^{2} \) |
| 47 | \( 1 + 5.76T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 + 8.88T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 - 9.59T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 + 5.89T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.281309550728136847840602518886, −7.68924718839764349945328810183, −6.62243099448065715812332948873, −6.02393363414844966553224566024, −5.13173687180765345443604641226, −4.82757493047801878413756294850, −3.72182836916553050431711000108, −2.26806311634257313015407009517, −1.53576914066282992121202055297, 0,
1.53576914066282992121202055297, 2.26806311634257313015407009517, 3.72182836916553050431711000108, 4.82757493047801878413756294850, 5.13173687180765345443604641226, 6.02393363414844966553224566024, 6.62243099448065715812332948873, 7.68924718839764349945328810183, 8.281309550728136847840602518886
