| L(s) = 1 | − 2.67·2-s − 3.22·3-s + 5.17·4-s − 5-s + 8.63·6-s − 7-s − 8.48·8-s + 7.38·9-s + 2.67·10-s + 0.611·11-s − 16.6·12-s + 1.44·13-s + 2.67·14-s + 3.22·15-s + 12.3·16-s + 0.0160·17-s − 19.7·18-s − 4.79·19-s − 5.17·20-s + 3.22·21-s − 1.63·22-s + 5.47·23-s + 27.3·24-s + 25-s − 3.86·26-s − 14.1·27-s − 5.17·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 1.86·3-s + 2.58·4-s − 0.447·5-s + 3.52·6-s − 0.377·7-s − 3.00·8-s + 2.46·9-s + 0.846·10-s + 0.184·11-s − 4.81·12-s + 0.400·13-s + 0.715·14-s + 0.832·15-s + 3.09·16-s + 0.00389·17-s − 4.66·18-s − 1.09·19-s − 1.15·20-s + 0.703·21-s − 0.349·22-s + 1.14·23-s + 5.58·24-s + 0.200·25-s − 0.758·26-s − 2.72·27-s − 0.977·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 + 3.22T + 3T^{2} \) |
| 11 | \( 1 - 0.611T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 - 0.0160T + 17T^{2} \) |
| 19 | \( 1 + 4.79T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 - 6.57T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 + 6.51T + 41T^{2} \) |
| 43 | \( 1 + 4.62T + 43T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 - 3.15T + 61T^{2} \) |
| 67 | \( 1 - 2.84T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 + 0.264T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 4.03T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29459527197908388424272642539, −6.82511542551088299726521009021, −6.49134308590209945075693116453, −5.74697089890403069688323965824, −4.89936491520723986653772981122, −3.92968127478733451631211738046, −2.77014749019747343018885093147, −1.55968360223466100394071840199, −0.829747218649252328207065472620, 0,
0.829747218649252328207065472620, 1.55968360223466100394071840199, 2.77014749019747343018885093147, 3.92968127478733451631211738046, 4.89936491520723986653772981122, 5.74697089890403069688323965824, 6.49134308590209945075693116453, 6.82511542551088299726521009021, 7.29459527197908388424272642539
