| L(s) = 1 | + 2-s + 0.334·3-s + 4-s + 2.22·5-s + 0.334·6-s + 7-s + 8-s − 2.88·9-s + 2.22·10-s − 0.585·11-s + 0.334·12-s − 5.97·13-s + 14-s + 0.744·15-s + 16-s − 4.22·17-s − 2.88·18-s − 2.66·19-s + 2.22·20-s + 0.334·21-s − 0.585·22-s + 0.334·24-s − 0.0594·25-s − 5.97·26-s − 1.97·27-s + 28-s − 4.43·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.193·3-s + 0.5·4-s + 0.994·5-s + 0.136·6-s + 0.377·7-s + 0.353·8-s − 0.962·9-s + 0.702·10-s − 0.176·11-s + 0.0966·12-s − 1.65·13-s + 0.267·14-s + 0.192·15-s + 0.250·16-s − 1.02·17-s − 0.680·18-s − 0.612·19-s + 0.497·20-s + 0.0730·21-s − 0.124·22-s + 0.0683·24-s − 0.0118·25-s − 1.17·26-s − 0.379·27-s + 0.188·28-s − 0.824·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.334T + 3T^{2} \) |
| 5 | \( 1 - 2.22T + 5T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 13 | \( 1 + 5.97T + 13T^{2} \) |
| 17 | \( 1 + 4.22T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 37 | \( 1 - 3.53T + 37T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 - 5.78T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 + 9.19T + 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 + 1.23T + 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 9.97T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−7.53301620579860632893146363840, −6.68626159935202102483096058950, −6.00566205008428040034943094338, −5.48005293245871832477808506968, −4.72050045029346929099808465857, −4.15789382128462550855632306443, −2.77844689125762301277152475386, −2.52373087113946499360204248169, −1.69314216198930349475120799800, 0,
1.69314216198930349475120799800, 2.52373087113946499360204248169, 2.77844689125762301277152475386, 4.15789382128462550855632306443, 4.72050045029346929099808465857, 5.48005293245871832477808506968, 6.00566205008428040034943094338, 6.68626159935202102483096058950, 7.53301620579860632893146363840
