| L(s) = 1 | + 16·2-s + 196.·3-s + 256·4-s + 3.14e3·6-s + 2.40e3·7-s + 4.09e3·8-s + 1.90e4·9-s − 4.77e4·11-s + 5.03e4·12-s − 1.81e5·13-s + 3.84e4·14-s + 6.55e4·16-s − 2.85e5·17-s + 3.04e5·18-s + 1.54e5·19-s + 4.72e5·21-s − 7.64e5·22-s + 8.04e5·23-s + 8.06e5·24-s − 2.89e6·26-s − 1.26e5·27-s + 6.14e5·28-s + 5.48e6·29-s − 9.13e6·31-s + 1.04e6·32-s − 9.40e6·33-s − 4.56e6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.40·3-s + 0.5·4-s + 0.991·6-s + 0.377·7-s + 0.353·8-s + 0.967·9-s − 0.984·11-s + 0.701·12-s − 1.75·13-s + 0.267·14-s + 0.250·16-s − 0.828·17-s + 0.684·18-s + 0.272·19-s + 0.530·21-s − 0.696·22-s + 0.599·23-s + 0.495·24-s − 1.24·26-s − 0.0457·27-s + 0.188·28-s + 1.44·29-s − 1.77·31-s + 0.176·32-s − 1.38·33-s − 0.585·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 - 196.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 4.77e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.81e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.85e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.54e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.04e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.48e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.13e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.11e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.57e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.20e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.71e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.33e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.15e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.49e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.15e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.48e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.71e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.48e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.68e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.49e8T + 7.60e17T^{2} \) |
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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
−9.431328918695942808181975775479, −8.429909578372592475719100407652, −7.61555074514243760823924841578, −6.89256078273068331242657596968, −5.26709785878304064101190215314, −4.60044384042829983943625674772, −3.30169110410260326222820402671, −2.56240101668785161904205567659, −1.81893325453965674797875617439, 0,
1.81893325453965674797875617439, 2.56240101668785161904205567659, 3.30169110410260326222820402671, 4.60044384042829983943625674772, 5.26709785878304064101190215314, 6.89256078273068331242657596968, 7.61555074514243760823924841578, 8.429909578372592475719100407652, 9.431328918695942808181975775479
