| L(s) = 1 | + 3-s + 0.747·7-s + 9-s + 0.209·11-s − 2.50·13-s − 6.81·17-s + 1.24·19-s + 0.747·21-s + 3.25·23-s + 27-s − 5.18·29-s + 3.73·31-s + 0.209·33-s + 1.96·37-s − 2.50·39-s + 3.21·41-s − 12.7·43-s − 6.48·47-s − 6.44·49-s − 6.81·51-s + 4.14·53-s + 1.24·57-s + 11.7·59-s − 12.4·61-s + 0.747·63-s + 2.91·67-s + 3.25·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.282·7-s + 0.333·9-s + 0.0630·11-s − 0.694·13-s − 1.65·17-s + 0.285·19-s + 0.163·21-s + 0.677·23-s + 0.192·27-s − 0.962·29-s + 0.671·31-s + 0.0363·33-s + 0.323·37-s − 0.400·39-s + 0.501·41-s − 1.93·43-s − 0.945·47-s − 0.920·49-s − 0.954·51-s + 0.569·53-s + 0.164·57-s + 1.52·59-s − 1.59·61-s + 0.0941·63-s + 0.356·67-s + 0.391·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.747T + 7T^{2} \) |
| 11 | \( 1 - 0.209T + 11T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 + 6.81T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 + 5.18T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 - 1.96T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 - 4.14T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.91T + 67T^{2} \) |
| 71 | \( 1 - 6.55T + 71T^{2} \) |
| 73 | \( 1 + 2.51T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 5.67T + 83T^{2} \) |
| 89 | \( 1 + 0.921T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.52898818325816741165894257820, −6.92172985148969834824134194067, −6.33390102424163834493797319202, −5.25034016916104295939772364538, −4.69214840936118128922981503160, −3.96078308231019895119552553792, −3.02673707695769526978634943027, −2.30440286563450660839587159180, −1.45663999495648893889550801483, 0,
1.45663999495648893889550801483, 2.30440286563450660839587159180, 3.02673707695769526978634943027, 3.96078308231019895119552553792, 4.69214840936118128922981503160, 5.25034016916104295939772364538, 6.33390102424163834493797319202, 6.92172985148969834824134194067, 7.52898818325816741165894257820
