L(s) = 1 | − 2-s − 2.89·3-s + 4-s + 2.32·5-s + 2.89·6-s + 0.619·7-s − 8-s + 5.36·9-s − 2.32·10-s − 11-s − 2.89·12-s − 5.40·13-s − 0.619·14-s − 6.71·15-s + 16-s − 2.11·17-s − 5.36·18-s + 2.32·20-s − 1.79·21-s + 22-s + 1.91·23-s + 2.89·24-s + 0.389·25-s + 5.40·26-s − 6.83·27-s + 0.619·28-s + 8.17·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.66·3-s + 0.5·4-s + 1.03·5-s + 1.18·6-s + 0.234·7-s − 0.353·8-s + 1.78·9-s − 0.734·10-s − 0.301·11-s − 0.834·12-s − 1.49·13-s − 0.165·14-s − 1.73·15-s + 0.250·16-s − 0.511·17-s − 1.26·18-s + 0.519·20-s − 0.391·21-s + 0.213·22-s + 0.399·23-s + 0.590·24-s + 0.0779·25-s + 1.06·26-s − 1.31·27-s + 0.117·28-s + 1.51·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 - 0.619T + 7T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 23 | \( 1 - 1.91T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 41 | \( 1 - 3.78T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 - 6.91T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 0.481T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 0.861T + 71T^{2} \) |
| 73 | \( 1 - 0.742T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 9.63T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 8.77T + 97T^{2} \) |
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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
−7.20833129687096225068137369458, −6.83327642766740145425205049090, −6.13323687721497451493481986605, −5.40577887945898165078048880435, −5.06001561399428859446247039435, −4.22027969823775315664861306045, −2.71500392316906545885466643781, −2.00829884091825367709147625389, −1.00928750771960211338301986773, 0,
1.00928750771960211338301986773, 2.00829884091825367709147625389, 2.71500392316906545885466643781, 4.22027969823775315664861306045, 5.06001561399428859446247039435, 5.40577887945898165078048880435, 6.13323687721497451493481986605, 6.83327642766740145425205049090, 7.20833129687096225068137369458