| L(s) = 1 | + 3.96·2-s + 3·3-s + 7.73·4-s − 5·5-s + 11.8·6-s − 32.8·7-s − 1.07·8-s + 9·9-s − 19.8·10-s + 12.6·11-s + 23.1·12-s − 52.2·13-s − 130.·14-s − 15·15-s − 66.0·16-s − 17.2·17-s + 35.6·18-s + 48.0·19-s − 38.6·20-s − 98.6·21-s + 50.1·22-s − 130.·23-s − 3.21·24-s + 25·25-s − 207.·26-s + 27·27-s − 254.·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.577·3-s + 0.966·4-s − 0.447·5-s + 0.809·6-s − 1.77·7-s − 0.0473·8-s + 0.333·9-s − 0.627·10-s + 0.346·11-s + 0.557·12-s − 1.11·13-s − 2.48·14-s − 0.258·15-s − 1.03·16-s − 0.246·17-s + 0.467·18-s + 0.580·19-s − 0.432·20-s − 1.02·21-s + 0.485·22-s − 1.18·23-s − 0.0273·24-s + 0.200·25-s − 1.56·26-s + 0.192·27-s − 1.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 3.96T + 8T^{2} \) |
| 7 | \( 1 + 32.8T + 343T^{2} \) |
| 11 | \( 1 - 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 134.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 222.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 569.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 393.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 741.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 758.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 74.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 118.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 498.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 106.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 42.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 161.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 128.T + 9.12e5T^{2} \) |
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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
−10.08262203894597119407069920106, −9.577906284821710044346902103929, −8.420423375750180586494432961170, −7.05439974327225007102034326056, −6.49401552665353551291449814700, −5.31141969719610588493107176905, −4.11616575816282889043831038620, −3.37804862897951705431615827181, −2.50698734870844176388196730566, 0,
2.50698734870844176388196730566, 3.37804862897951705431615827181, 4.11616575816282889043831038620, 5.31141969719610588493107176905, 6.49401552665353551291449814700, 7.05439974327225007102034326056, 8.420423375750180586494432961170, 9.577906284821710044346902103929, 10.08262203894597119407069920106
