L(s) = 1 | + 2-s − 2.39·3-s + 4-s − 3.46·5-s − 2.39·6-s − 3.11·7-s + 8-s + 2.73·9-s − 3.46·10-s − 4.76·11-s − 2.39·12-s − 3.60·13-s − 3.11·14-s + 8.30·15-s + 16-s + 2.91·17-s + 2.73·18-s + 19-s − 3.46·20-s + 7.46·21-s − 4.76·22-s + 4.74·23-s − 2.39·24-s + 7.01·25-s − 3.60·26-s + 0.624·27-s − 3.11·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.38·3-s + 0.5·4-s − 1.55·5-s − 0.978·6-s − 1.17·7-s + 0.353·8-s + 0.913·9-s − 1.09·10-s − 1.43·11-s − 0.691·12-s − 0.998·13-s − 0.832·14-s + 2.14·15-s + 0.250·16-s + 0.707·17-s + 0.645·18-s + 0.229·19-s − 0.775·20-s + 1.62·21-s − 1.01·22-s + 0.989·23-s − 0.489·24-s + 1.40·25-s − 0.706·26-s + 0.120·27-s − 0.588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 2.39T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 4.17T + 29T^{2} \) |
| 31 | \( 1 - 6.99T + 31T^{2} \) |
| 37 | \( 1 - 0.716T + 37T^{2} \) |
| 41 | \( 1 + 3.08T + 41T^{2} \) |
| 43 | \( 1 - 0.576T + 43T^{2} \) |
| 47 | \( 1 - 1.44T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 1.24T + 61T^{2} \) |
| 67 | \( 1 + 0.735T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 5.48T + 73T^{2} \) |
| 79 | \( 1 + 3.39T + 79T^{2} \) |
| 83 | \( 1 + 9.99T + 83T^{2} \) |
| 89 | \( 1 - 6.54T + 89T^{2} \) |
| 97 | \( 1 - 3.66T + 97T^{2} \) |
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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
−7.22885834302014696442543557832, −6.87092865816249762480284995016, −5.92414372918471504508804057226, −5.32654658221874643514780511694, −4.81209850671009207866682662981, −4.04748122770274796378839457771, −3.17593509897632323641492360907, −2.64395039297522240861736836196, −0.77475302560310758174189215675, 0,
0.77475302560310758174189215675, 2.64395039297522240861736836196, 3.17593509897632323641492360907, 4.04748122770274796378839457771, 4.81209850671009207866682662981, 5.32654658221874643514780511694, 5.92414372918471504508804057226, 6.87092865816249762480284995016, 7.22885834302014696442543557832