| L(s) = 1 | − 2-s + 2.84·3-s + 4-s − 0.450·5-s − 2.84·6-s − 7-s − 8-s + 5.08·9-s + 0.450·10-s − 3.81·11-s + 2.84·12-s + 5.33·13-s + 14-s − 1.28·15-s + 16-s − 5.31·17-s − 5.08·18-s − 6.32·19-s − 0.450·20-s − 2.84·21-s + 3.81·22-s − 9.02·23-s − 2.84·24-s − 4.79·25-s − 5.33·26-s + 5.92·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.64·3-s + 0.5·4-s − 0.201·5-s − 1.16·6-s − 0.377·7-s − 0.353·8-s + 1.69·9-s + 0.142·10-s − 1.14·11-s + 0.820·12-s + 1.47·13-s + 0.267·14-s − 0.330·15-s + 0.250·16-s − 1.29·17-s − 1.19·18-s − 1.45·19-s − 0.100·20-s − 0.620·21-s + 0.812·22-s − 1.88·23-s − 0.580·24-s − 0.959·25-s − 1.04·26-s + 1.13·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.84T + 3T^{2} \) |
| 5 | \( 1 + 0.450T + 5T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 + 9.02T + 23T^{2} \) |
| 29 | \( 1 - 8.31T + 29T^{2} \) |
| 31 | \( 1 - 7.67T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 - 0.836T + 53T^{2} \) |
| 59 | \( 1 + 1.36T + 59T^{2} \) |
| 61 | \( 1 + 7.00T + 61T^{2} \) |
| 67 | \( 1 + 8.41T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 0.345T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 6.49T + 89T^{2} \) |
| 97 | \( 1 + 6.29T + 97T^{2} \) |
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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
−8.026587777683710731010730606394, −7.37485600958450964459751667817, −6.35001760054917897962399194281, −5.99813769626141000090254956201, −4.22312375268846403631962992257, −4.13938520080905723301336910815, −2.81561939900594425211294902330, −2.50102224905906910089580848984, −1.54355666416194891483949663997, 0,
1.54355666416194891483949663997, 2.50102224905906910089580848984, 2.81561939900594425211294902330, 4.13938520080905723301336910815, 4.22312375268846403631962992257, 5.99813769626141000090254956201, 6.35001760054917897962399194281, 7.37485600958450964459751667817, 8.026587777683710731010730606394
