| L(s) = 1 | + 0.414·2-s + 1.41·3-s − 1.82·4-s − 5-s + 0.585·6-s − 1.58·8-s − 0.999·9-s − 0.414·10-s + 2.41·11-s − 2.58·12-s − 6.24·13-s − 1.41·15-s + 3·16-s + 4·17-s − 0.414·18-s + 19-s + 1.82·20-s + 0.999·22-s − 8.41·23-s − 2.24·24-s − 4·25-s − 2.58·26-s − 5.65·27-s − 9.41·29-s − 0.585·30-s − 2.24·31-s + 4.41·32-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + 0.816·3-s − 0.914·4-s − 0.447·5-s + 0.239·6-s − 0.560·8-s − 0.333·9-s − 0.130·10-s + 0.727·11-s − 0.746·12-s − 1.73·13-s − 0.365·15-s + 0.750·16-s + 0.970·17-s − 0.0976·18-s + 0.229·19-s + 0.408·20-s + 0.213·22-s − 1.75·23-s − 0.457·24-s − 0.800·25-s − 0.507·26-s − 1.08·27-s − 1.74·29-s − 0.106·30-s − 0.402·31-s + 0.780·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 + 8.41T + 23T^{2} \) |
| 29 | \( 1 + 9.41T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 + 6.17T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 + 5.75T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + 0.757T + 83T^{2} \) |
| 89 | \( 1 - 8.58T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−9.541190086286756481127264913467, −8.919350983339019481553654851602, −7.84756790159985020137769424505, −7.54716033439586061452896753301, −5.96991597778362761330439416335, −5.17525280820052133400766889540, −4.00353866963766199997818590725, −3.46245215527175772849937367516, −2.13298614563174457162592794819, 0,
2.13298614563174457162592794819, 3.46245215527175772849937367516, 4.00353866963766199997818590725, 5.17525280820052133400766889540, 5.96991597778362761330439416335, 7.54716033439586061452896753301, 7.84756790159985020137769424505, 8.919350983339019481553654851602, 9.541190086286756481127264913467
