| L(s) = 1 | − 2-s + 3-s + 4-s + 1.79·5-s − 6-s − 8-s + 9-s − 1.79·10-s − 2.15·11-s + 12-s − 1.79·13-s + 1.79·15-s + 16-s − 5.79·17-s − 18-s + 19-s + 1.79·20-s + 2.15·22-s + 0.383·23-s − 24-s − 1.76·25-s + 1.79·26-s + 27-s + 0.317·29-s − 1.79·30-s + 7.37·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.804·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.568·10-s − 0.650·11-s + 0.288·12-s − 0.498·13-s + 0.464·15-s + 0.250·16-s − 1.40·17-s − 0.235·18-s + 0.229·19-s + 0.402·20-s + 0.460·22-s + 0.0800·23-s − 0.204·24-s − 0.353·25-s + 0.352·26-s + 0.192·27-s + 0.0590·29-s − 0.328·30-s + 1.32·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 23 | \( 1 - 0.383T + 23T^{2} \) |
| 29 | \( 1 - 0.317T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 + 8.10T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 - 7.97T + 53T^{2} \) |
| 59 | \( 1 + 5.08T + 59T^{2} \) |
| 61 | \( 1 + 0.810T + 61T^{2} \) |
| 67 | \( 1 - 5.72T + 67T^{2} \) |
| 71 | \( 1 - 3.91T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 6.28T + 97T^{2} \) |
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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
−7.987257538586694479574231501423, −7.02656952334446331275286387173, −6.67674572555106552195208757493, −5.66983672540606147815268555206, −4.97268616974930485923743893819, −4.01409245184586728074699062250, −2.86778051292510838045449327225, −2.31471813794386560553759639240, −1.49339041458374368705500578207, 0,
1.49339041458374368705500578207, 2.31471813794386560553759639240, 2.86778051292510838045449327225, 4.01409245184586728074699062250, 4.97268616974930485923743893819, 5.66983672540606147815268555206, 6.67674572555106552195208757493, 7.02656952334446331275286387173, 7.987257538586694479574231501423
