| L(s) = 1 | − 0.656·2-s − 1.56·4-s − 1.34·7-s + 2.34·8-s − 4.16·11-s − 3.56·13-s + 0.882·14-s + 1.59·16-s + 0.744·17-s − 1.74·19-s + 2.73·22-s + 4.25·23-s + 2.34·26-s + 2.10·28-s − 1.68·29-s + 8.13·31-s − 5.73·32-s − 0.488·34-s − 37-s + 1.14·38-s + 11.6·41-s − 8.39·43-s + 6.53·44-s − 2.79·46-s + 8.79·47-s − 5.19·49-s + 5.59·52-s + ⋯ |
| L(s) = 1 | − 0.464·2-s − 0.784·4-s − 0.507·7-s + 0.828·8-s − 1.25·11-s − 0.989·13-s + 0.235·14-s + 0.399·16-s + 0.180·17-s − 0.400·19-s + 0.583·22-s + 0.887·23-s + 0.459·26-s + 0.398·28-s − 0.313·29-s + 1.46·31-s − 1.01·32-s − 0.0838·34-s − 0.164·37-s + 0.185·38-s + 1.82·41-s − 1.27·43-s + 0.985·44-s − 0.411·46-s + 1.28·47-s − 0.742·49-s + 0.776·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.656T + 2T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 0.744T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 31 | \( 1 - 8.13T + 31T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 8.39T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 - 1.48T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.59T + 61T^{2} \) |
| 67 | \( 1 - 4.28T + 67T^{2} \) |
| 71 | \( 1 - 1.70T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 - 7.56T + 83T^{2} \) |
| 89 | \( 1 + 5.37T + 89T^{2} \) |
| 97 | \( 1 - 1.13T + 97T^{2} \) |
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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
−7.57260327553159590832016423342, −6.98492143526365939775715279967, −6.03952960387302177682255679538, −5.21251413801923498693531853228, −4.77897968702262089232936856884, −3.95061875878083724948759159853, −2.96689293307504430123651100023, −2.29493719798171049368026411605, −0.931260087716432175170713519188, 0,
0.931260087716432175170713519188, 2.29493719798171049368026411605, 2.96689293307504430123651100023, 3.95061875878083724948759159853, 4.77897968702262089232936856884, 5.21251413801923498693531853228, 6.03952960387302177682255679538, 6.98492143526365939775715279967, 7.57260327553159590832016423342
