| L(s) = 1 | − 10.9·2-s − 11.8·3-s + 87.9·4-s + 129.·6-s + 32.4·7-s − 613.·8-s − 103.·9-s + 452.·11-s − 1.03e3·12-s + 141.·13-s − 355.·14-s + 3.89e3·16-s − 675.·17-s + 1.13e3·18-s + 2.39e3·19-s − 383.·21-s − 4.95e3·22-s + 555.·23-s + 7.24e3·24-s − 1.55e3·26-s + 4.09e3·27-s + 2.85e3·28-s + 3.18e3·29-s − 5.90e3·31-s − 2.30e4·32-s − 5.34e3·33-s + 7.39e3·34-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.757·3-s + 2.74·4-s + 1.46·6-s + 0.250·7-s − 3.38·8-s − 0.425·9-s + 1.12·11-s − 2.08·12-s + 0.232·13-s − 0.484·14-s + 3.80·16-s − 0.566·17-s + 0.824·18-s + 1.51·19-s − 0.189·21-s − 2.18·22-s + 0.219·23-s + 2.56·24-s − 0.450·26-s + 1.08·27-s + 0.688·28-s + 0.704·29-s − 1.10·31-s − 3.98·32-s − 0.854·33-s + 1.09·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7377200190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7377200190\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 10.9T + 32T^{2} \) |
| 3 | \( 1 + 11.8T + 243T^{2} \) |
| 7 | \( 1 - 32.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 452.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 141.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 675.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.39e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 555.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.47e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 1.75e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.27e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.51e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.80e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.52e5T + 8.58e9T^{2} \) |
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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
−9.136840790869739148390416848232, −8.556765816412987408209989508082, −7.58993537872908893457327113905, −6.83940803007399237163833614548, −6.15204791692031286452288133488, −5.25579315532450992715748941667, −3.56127303701466060428112519211, −2.38273445128117543397349970545, −1.23816088739705343414513877680, −0.59818310591473702178883931042,
0.59818310591473702178883931042, 1.23816088739705343414513877680, 2.38273445128117543397349970545, 3.56127303701466060428112519211, 5.25579315532450992715748941667, 6.15204791692031286452288133488, 6.83940803007399237163833614548, 7.58993537872908893457327113905, 8.556765816412987408209989508082, 9.136840790869739148390416848232
