| L(s) = 1 | + 2.76·2-s − 2.98·3-s + 5.62·4-s + 3.27·5-s − 8.25·6-s + 10.0·8-s + 5.93·9-s + 9.05·10-s + 11-s − 16.8·12-s + 13-s − 9.79·15-s + 16.3·16-s + 4.19·17-s + 16.3·18-s − 7.74·19-s + 18.4·20-s + 2.76·22-s − 2.32·23-s − 29.8·24-s + 5.74·25-s + 2.76·26-s − 8.76·27-s + 6.38·29-s − 27.0·30-s + 4.72·31-s + 25.2·32-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 1.72·3-s + 2.81·4-s + 1.46·5-s − 3.36·6-s + 3.53·8-s + 1.97·9-s + 2.86·10-s + 0.301·11-s − 4.85·12-s + 0.277·13-s − 2.52·15-s + 4.09·16-s + 1.01·17-s + 3.86·18-s − 1.77·19-s + 4.12·20-s + 0.588·22-s − 0.485·23-s − 6.10·24-s + 1.14·25-s + 0.541·26-s − 1.68·27-s + 1.18·29-s − 4.93·30-s + 0.848·31-s + 4.45·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.592676430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.592676430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 7.74T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 - 4.72T + 31T^{2} \) |
| 37 | \( 1 - 1.92T + 37T^{2} \) |
| 41 | \( 1 + 0.0990T + 41T^{2} \) |
| 43 | \( 1 + 0.570T + 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 - 0.784T + 61T^{2} \) |
| 67 | \( 1 + 4.93T + 67T^{2} \) |
| 71 | \( 1 + 3.90T + 71T^{2} \) |
| 73 | \( 1 + 8.28T + 73T^{2} \) |
| 79 | \( 1 + 9.54T + 79T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 + 0.293T + 89T^{2} \) |
| 97 | \( 1 + 8.10T + 97T^{2} \) |
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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
−7.34892123943771631131168548167, −6.53508961475467714221966587630, −6.19578071032434588379579698446, −5.92666974995829214661980067287, −5.14987845676225891590015270886, −4.64861979538624349338608006295, −3.96620256181864409105396933128, −2.82035131568881364307402722547, −1.91185367743839638808335922059, −1.16902929888065573888742721636,
1.16902929888065573888742721636, 1.91185367743839638808335922059, 2.82035131568881364307402722547, 3.96620256181864409105396933128, 4.64861979538624349338608006295, 5.14987845676225891590015270886, 5.92666974995829214661980067287, 6.19578071032434588379579698446, 6.53508961475467714221966587630, 7.34892123943771631131168548167
