L(s) = 1 | − 1.82·2-s + 3-s + 1.31·4-s + 3.98·5-s − 1.82·6-s − 0.917·7-s + 1.24·8-s + 9-s − 7.25·10-s − 5.15·11-s + 1.31·12-s + 13-s + 1.67·14-s + 3.98·15-s − 4.89·16-s + 1.23·17-s − 1.82·18-s − 6.69·19-s + 5.24·20-s − 0.917·21-s + 9.38·22-s − 8.21·23-s + 1.24·24-s + 10.8·25-s − 1.82·26-s + 27-s − 1.20·28-s + ⋯ |
L(s) = 1 | − 1.28·2-s + 0.577·3-s + 0.658·4-s + 1.78·5-s − 0.743·6-s − 0.346·7-s + 0.439·8-s + 0.333·9-s − 2.29·10-s − 1.55·11-s + 0.380·12-s + 0.277·13-s + 0.446·14-s + 1.02·15-s − 1.22·16-s + 0.300·17-s − 0.429·18-s − 1.53·19-s + 1.17·20-s − 0.200·21-s + 2.00·22-s − 1.71·23-s + 0.253·24-s + 2.17·25-s − 0.357·26-s + 0.192·27-s − 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 5 | \( 1 - 3.98T + 5T^{2} \) |
| 7 | \( 1 + 0.917T + 7T^{2} \) |
| 11 | \( 1 + 5.15T + 11T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 + 6.69T + 19T^{2} \) |
| 23 | \( 1 + 8.21T + 23T^{2} \) |
| 29 | \( 1 - 3.31T + 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 4.75T + 43T^{2} \) |
| 47 | \( 1 + 5.00T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 - 0.0713T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 4.96T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 + 2.17T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.270932789782011459526010646008, −7.69731797772586080712255861026, −6.63844952147937331344642662192, −6.10660297561899753141049860529, −5.22198924393181685857293043401, −4.31285181305659347561339819495, −2.88789382030685482393723650221, −2.18757461795782009641714441318, −1.55520491140217170338827207111, 0,
1.55520491140217170338827207111, 2.18757461795782009641714441318, 2.88789382030685482393723650221, 4.31285181305659347561339819495, 5.22198924393181685857293043401, 6.10660297561899753141049860529, 6.63844952147937331344642662192, 7.69731797772586080712255861026, 8.270932789782011459526010646008