| L(s) = 1 | + 3-s + 2.47·7-s + 9-s − 3.58·11-s − 5.36·13-s − 3.43·17-s + 7.10·19-s + 2.47·21-s + 7.54·23-s + 27-s + 1.11·29-s − 31-s − 3.58·33-s + 10.2·37-s − 5.36·39-s + 6.88·41-s + 1.70·43-s − 9.03·47-s − 0.850·49-s − 3.43·51-s − 10.2·53-s + 7.10·57-s + 0.287·59-s − 5.12·61-s + 2.47·63-s + 4.49·67-s + 7.54·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.937·7-s + 0.333·9-s − 1.08·11-s − 1.48·13-s − 0.833·17-s + 1.63·19-s + 0.541·21-s + 1.57·23-s + 0.192·27-s + 0.207·29-s − 0.179·31-s − 0.623·33-s + 1.67·37-s − 0.858·39-s + 1.07·41-s + 0.259·43-s − 1.31·47-s − 0.121·49-s − 0.481·51-s − 1.40·53-s + 0.941·57-s + 0.0373·59-s − 0.656·61-s + 0.312·63-s + 0.549·67-s + 0.908·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.633253375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.633253375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2.47T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 - 1.11T + 29T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 6.88T + 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 0.287T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 - 4.49T + 67T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8.37T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 5.44T + 89T^{2} \) |
| 97 | \( 1 + 5.33T + 97T^{2} \) |
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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
−7.77708685755554750093214623552, −7.30062092166922431314353010792, −6.48676053190966101971572685335, −5.36307993967165515359264001908, −4.91198598398583670655778939057, −4.45395505439645985785454362976, −3.16226453768566482259655144699, −2.68828441977771839435038588276, −1.88425688563469183345856028571, −0.75325516685320026180198724316,
0.75325516685320026180198724316, 1.88425688563469183345856028571, 2.68828441977771839435038588276, 3.16226453768566482259655144699, 4.45395505439645985785454362976, 4.91198598398583670655778939057, 5.36307993967165515359264001908, 6.48676053190966101971572685335, 7.30062092166922431314353010792, 7.77708685755554750093214623552
