L(s) = 1 | + 1.29·2-s − 0.310·4-s − 3.52·5-s − 3.00·8-s − 4.58·10-s + 1.17·11-s + 3.22·13-s − 3.28·16-s + 4.90·17-s + 6.86·19-s + 1.09·20-s + 1.53·22-s − 4.29·23-s + 7.43·25-s + 4.18·26-s − 2.72·29-s + 1.92·31-s + 1.73·32-s + 6.37·34-s − 9.76·37-s + 8.92·38-s + 10.5·40-s − 6.65·41-s − 9.66·43-s − 0.365·44-s − 5.58·46-s − 0.633·47-s + ⋯ |
L(s) = 1 | + 0.919·2-s − 0.155·4-s − 1.57·5-s − 1.06·8-s − 1.44·10-s + 0.355·11-s + 0.893·13-s − 0.820·16-s + 1.18·17-s + 1.57·19-s + 0.244·20-s + 0.326·22-s − 0.896·23-s + 1.48·25-s + 0.821·26-s − 0.505·29-s + 0.344·31-s + 0.307·32-s + 1.09·34-s − 1.60·37-s + 1.44·38-s + 1.67·40-s − 1.03·41-s − 1.47·43-s − 0.0551·44-s − 0.824·46-s − 0.0923·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 - 4.90T + 17T^{2} \) |
| 19 | \( 1 - 6.86T + 19T^{2} \) |
| 23 | \( 1 + 4.29T + 23T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 - 1.92T + 31T^{2} \) |
| 37 | \( 1 + 9.76T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 + 9.66T + 43T^{2} \) |
| 47 | \( 1 + 0.633T + 47T^{2} \) |
| 53 | \( 1 + 2.22T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 5.33T + 67T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 + 1.03T + 73T^{2} \) |
| 79 | \( 1 - 1.00T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.168398249533081053234038607868, −7.33133659017985148021995277846, −6.58571544792468965189863105902, −5.58874653578254762275712836982, −5.05446938727318218037039699328, −4.08718499046633238316133920371, −3.52453102780798530416122355370, −3.16092783012639919718146419418, −1.32100027347833208105545051225, 0,
1.32100027347833208105545051225, 3.16092783012639919718146419418, 3.52453102780798530416122355370, 4.08718499046633238316133920371, 5.05446938727318218037039699328, 5.58874653578254762275712836982, 6.58571544792468965189863105902, 7.33133659017985148021995277846, 8.168398249533081053234038607868