| L(s) = 1 | − 2.02·2-s + 0.680·3-s + 2.10·4-s + 5-s − 1.37·6-s + 7-s − 0.213·8-s − 2.53·9-s − 2.02·10-s + 2.03·11-s + 1.43·12-s − 5.76·13-s − 2.02·14-s + 0.680·15-s − 3.77·16-s − 1.08·17-s + 5.13·18-s + 1.79·19-s + 2.10·20-s + 0.680·21-s − 4.13·22-s − 5.84·23-s − 0.145·24-s + 25-s + 11.6·26-s − 3.76·27-s + 2.10·28-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 0.392·3-s + 1.05·4-s + 0.447·5-s − 0.562·6-s + 0.377·7-s − 0.0754·8-s − 0.845·9-s − 0.640·10-s + 0.614·11-s + 0.413·12-s − 1.59·13-s − 0.541·14-s + 0.175·15-s − 0.944·16-s − 0.262·17-s + 1.21·18-s + 0.412·19-s + 0.470·20-s + 0.148·21-s − 0.880·22-s − 1.21·23-s − 0.0296·24-s + 0.200·25-s + 2.29·26-s − 0.725·27-s + 0.397·28-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 | \( 1 + 2.02T + 2T^{2} \) |
| 3 | \( 1 - 0.680T + 3T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 5.76T + 13T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 - 9.45T + 37T^{2} \) |
| 41 | \( 1 + 6.19T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 5.99T + 47T^{2} \) |
| 53 | \( 1 - 3.69T + 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 + 2.59T + 67T^{2} \) |
| 71 | \( 1 - 0.698T + 71T^{2} \) |
| 73 | \( 1 - 2.68T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 1.17T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 1.33T + 97T^{2} \) |
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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
−7.64763687631420379925638109601, −7.16302999316804950322670813947, −6.29525651183401180028973396911, −5.52313593474310730229067883587, −4.69376787743652326580346154848, −3.83722115776456992894263499632, −2.49020249823607600972518276457, −2.27256621229120056742219848756, −1.11946297488148777846007728208, 0,
1.11946297488148777846007728208, 2.27256621229120056742219848756, 2.49020249823607600972518276457, 3.83722115776456992894263499632, 4.69376787743652326580346154848, 5.52313593474310730229067883587, 6.29525651183401180028973396911, 7.16302999316804950322670813947, 7.64763687631420379925638109601
