| L(s) = 1 | − 2.39·2-s + 3.71·4-s − 5-s − 1.32·7-s − 4.11·8-s + 2.39·10-s − 3·11-s + 6.11·13-s + 3.17·14-s + 2.39·16-s − 0.672·17-s − 5.43·19-s − 3.71·20-s + 7.17·22-s + 7.89·23-s + 25-s − 14.6·26-s − 4.93·28-s + 29-s + 2.50·32-s + 1.60·34-s + 1.32·35-s − 3.89·37-s + 13.0·38-s + 4.11·40-s − 4.32·41-s − 8.45·43-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.85·4-s − 0.447·5-s − 0.501·7-s − 1.45·8-s + 0.756·10-s − 0.904·11-s + 1.69·13-s + 0.848·14-s + 0.597·16-s − 0.163·17-s − 1.24·19-s − 0.831·20-s + 1.52·22-s + 1.64·23-s + 0.200·25-s − 2.86·26-s − 0.932·28-s + 0.185·29-s + 0.442·32-s + 0.275·34-s + 0.224·35-s − 0.639·37-s + 2.10·38-s + 0.649·40-s − 0.675·41-s − 1.28·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 7 | \( 1 + 1.32T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 + 0.672T + 17T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 + 4.32T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 3.21T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 + 1.10T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 9.20T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 0.889T + 83T^{2} \) |
| 89 | \( 1 - 2.33T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−9.016433213708273535051681906420, −8.528084431409831527464243263096, −7.973763008844740036931959743904, −6.86428818582604089364958964849, −6.46448378708401534638946222781, −5.14625513316087359525840431025, −3.73848039120127774903779037673, −2.67682380694961786233761682114, −1.34718162659656089720386186080, 0,
1.34718162659656089720386186080, 2.67682380694961786233761682114, 3.73848039120127774903779037673, 5.14625513316087359525840431025, 6.46448378708401534638946222781, 6.86428818582604089364958964849, 7.973763008844740036931959743904, 8.528084431409831527464243263096, 9.016433213708273535051681906420
