| L(s) = 1 | + 1.36·2-s − 2.64·3-s − 0.140·4-s − 5-s − 3.60·6-s + 1.30·7-s − 2.91·8-s + 3.98·9-s − 1.36·10-s + 11-s + 0.371·12-s − 3.57·13-s + 1.78·14-s + 2.64·15-s − 3.69·16-s + 5.39·17-s + 5.42·18-s + 0.902·19-s + 0.140·20-s − 3.45·21-s + 1.36·22-s − 1.29·23-s + 7.71·24-s + 25-s − 4.87·26-s − 2.58·27-s − 0.184·28-s + ⋯ |
| L(s) = 1 | + 0.964·2-s − 1.52·3-s − 0.0702·4-s − 0.447·5-s − 1.47·6-s + 0.494·7-s − 1.03·8-s + 1.32·9-s − 0.431·10-s + 0.301·11-s + 0.107·12-s − 0.991·13-s + 0.477·14-s + 0.682·15-s − 0.924·16-s + 1.30·17-s + 1.27·18-s + 0.207·19-s + 0.0314·20-s − 0.754·21-s + 0.290·22-s − 0.269·23-s + 1.57·24-s + 0.200·25-s − 0.956·26-s − 0.498·27-s − 0.0347·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 3 | \( 1 + 2.64T + 3T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 13 | \( 1 + 3.57T + 13T^{2} \) |
| 17 | \( 1 - 5.39T + 17T^{2} \) |
| 19 | \( 1 - 0.902T + 19T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 - 0.467T + 29T^{2} \) |
| 31 | \( 1 - 0.169T + 31T^{2} \) |
| 37 | \( 1 + 0.509T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 - 8.77T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 + 5.48T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 79 | \( 1 - 7.86T + 79T^{2} \) |
| 83 | \( 1 + 7.51T + 83T^{2} \) |
| 89 | \( 1 - 6.21T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−7.83291864234310578356406702682, −7.14929922948787983166598937493, −6.31870107336361891499993145247, −5.57377043027762945058273247645, −5.16580387111368564059890819196, −4.47512222580419342436203299399, −3.79150036685641084499328576734, −2.71467850289774271576056159910, −1.15934672077715866143055919677, 0,
1.15934672077715866143055919677, 2.71467850289774271576056159910, 3.79150036685641084499328576734, 4.47512222580419342436203299399, 5.16580387111368564059890819196, 5.57377043027762945058273247645, 6.31870107336361891499993145247, 7.14929922948787983166598937493, 7.83291864234310578356406702682
