| L(s) = 1 | − 0.688·2-s − 1.52·4-s − 4.90·7-s + 2.42·8-s + 3.21·11-s + 0.836·13-s + 3.37·14-s + 1.37·16-s − 6.73·17-s + 3·19-s − 2.21·22-s − 5.21·23-s − 0.576·26-s + 7.47·28-s − 1.49·29-s + 31-s − 5.80·32-s + 4.64·34-s − 0.407·37-s − 2.06·38-s + 9.54·41-s + 7.61·43-s − 4.90·44-s + 3.59·46-s + 6.62·47-s + 17.0·49-s − 1.27·52-s + ⋯ |
| L(s) = 1 | − 0.487·2-s − 0.762·4-s − 1.85·7-s + 0.858·8-s + 0.969·11-s + 0.232·13-s + 0.902·14-s + 0.344·16-s − 1.63·17-s + 0.688·19-s − 0.472·22-s − 1.08·23-s − 0.113·26-s + 1.41·28-s − 0.277·29-s + 0.179·31-s − 1.02·32-s + 0.796·34-s − 0.0670·37-s − 0.335·38-s + 1.49·41-s + 1.16·43-s − 0.739·44-s + 0.529·46-s + 0.965·47-s + 2.43·49-s − 0.176·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.688T + 2T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 13 | \( 1 - 0.836T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 37 | \( 1 + 0.407T + 37T^{2} \) |
| 41 | \( 1 - 9.54T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 - 6.62T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 - 0.755T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 + 8.29T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 2.78T + 89T^{2} \) |
| 97 | \( 1 + 8.95T + 97T^{2} \) |
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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
−7.41851467524818787739994097884, −7.10275674714411160090965968612, −5.99300811871781195143689308922, −5.91838812726611440861010091046, −4.39126688501447952312176769629, −4.09502241006624557611857846530, −3.24923533117135754815551102809, −2.25939636565397108095161824300, −0.952143551060639675623784722284, 0,
0.952143551060639675623784722284, 2.25939636565397108095161824300, 3.24923533117135754815551102809, 4.09502241006624557611857846530, 4.39126688501447952312176769629, 5.91838812726611440861010091046, 5.99300811871781195143689308922, 7.10275674714411160090965968612, 7.41851467524818787739994097884
