| L(s) = 1 | − 1.80·2-s + 3-s + 1.24·4-s − 5-s − 1.80·6-s + 2.80·7-s + 1.35·8-s + 9-s + 1.80·10-s + 3.49·11-s + 1.24·12-s − 5.04·14-s − 15-s − 4.93·16-s − 7.60·17-s − 1.80·18-s + 1.75·19-s − 1.24·20-s + 2.80·21-s − 6.29·22-s − 6.44·23-s + 1.35·24-s + 25-s + 27-s + 3.49·28-s − 9.74·29-s + 1.80·30-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.623·4-s − 0.447·5-s − 0.735·6-s + 1.05·7-s + 0.479·8-s + 0.333·9-s + 0.569·10-s + 1.05·11-s + 0.359·12-s − 1.34·14-s − 0.258·15-s − 1.23·16-s − 1.84·17-s − 0.424·18-s + 0.402·19-s − 0.278·20-s + 0.611·21-s − 1.34·22-s − 1.34·23-s + 0.276·24-s + 0.200·25-s + 0.192·27-s + 0.660·28-s − 1.80·29-s + 0.328·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 + 9.74T + 29T^{2} \) |
| 31 | \( 1 + 9.59T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 - 4.00T + 43T^{2} \) |
| 47 | \( 1 + 2.97T + 47T^{2} \) |
| 53 | \( 1 - 4.51T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 4.43T + 61T^{2} \) |
| 67 | \( 1 + 2.45T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 1.75T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 - 2.32T + 97T^{2} \) |
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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
−8.619073142407194497215766797107, −7.953581274842170924514234964575, −7.31434219702273036526518342219, −6.67082930258912340249350863303, −5.34662675583794823025150126557, −4.28413405190741872460227047411, −3.78978090977089315611512289330, −2.08541921595419328049380585429, −1.59779431297846226125231121597, 0,
1.59779431297846226125231121597, 2.08541921595419328049380585429, 3.78978090977089315611512289330, 4.28413405190741872460227047411, 5.34662675583794823025150126557, 6.67082930258912340249350863303, 7.31434219702273036526518342219, 7.953581274842170924514234964575, 8.619073142407194497215766797107
