| L(s) = 1 | − 1.23·2-s − 0.471·4-s − 3.27·7-s + 3.05·8-s − 2.30·11-s + 5.57·13-s + 4.05·14-s − 2.83·16-s + 1.94·17-s − 3.59·19-s + 2.84·22-s + 1.66·23-s − 6.89·26-s + 1.54·28-s − 29-s + 1.70·31-s − 2.60·32-s − 2.39·34-s − 9.16·37-s + 4.44·38-s + 8.54·41-s − 3.56·43-s + 1.08·44-s − 2.05·46-s − 11.5·47-s + 3.73·49-s − 2.62·52-s + ⋯ |
| L(s) = 1 | − 0.874·2-s − 0.235·4-s − 1.23·7-s + 1.08·8-s − 0.694·11-s + 1.54·13-s + 1.08·14-s − 0.708·16-s + 0.470·17-s − 0.823·19-s + 0.606·22-s + 0.346·23-s − 1.35·26-s + 0.291·28-s − 0.185·29-s + 0.306·31-s − 0.460·32-s − 0.411·34-s − 1.50·37-s + 0.720·38-s + 1.33·41-s − 0.543·43-s + 0.163·44-s − 0.303·46-s − 1.68·47-s + 0.533·49-s − 0.364·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 7 | \( 1 + 3.27T + 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 31 | \( 1 - 1.70T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 - 8.54T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 - 9.83T + 59T^{2} \) |
| 61 | \( 1 - 5.42T + 61T^{2} \) |
| 67 | \( 1 - 5.20T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 + 5.89T + 97T^{2} \) |
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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
−8.015148250592183757793907886141, −6.91098101913074315099803991585, −6.47277597439704426092090931430, −5.60728076536332471115212359258, −4.82544504531333511034320019172, −3.78100484164590848274556824765, −3.31576338829723681025399213314, −2.11055156903839115512337914651, −1.01212440022278897712629764799, 0,
1.01212440022278897712629764799, 2.11055156903839115512337914651, 3.31576338829723681025399213314, 3.78100484164590848274556824765, 4.82544504531333511034320019172, 5.60728076536332471115212359258, 6.47277597439704426092090931430, 6.91098101913074315099803991585, 8.015148250592183757793907886141
