| L(s) = 1 | + 0.414·3-s − 0.414·7-s − 2.82·9-s + 1.41·11-s − 3.82·13-s + 17-s + 19-s − 0.171·21-s + 3.24·23-s − 2.41·27-s − 1.82·29-s − 0.585·31-s + 0.585·33-s + 10.8·37-s − 1.58·39-s + 7.07·41-s + 6.24·43-s − 8·47-s − 6.82·49-s + 0.414·51-s + 3.82·53-s + 0.414·57-s − 11.5·59-s + 0.585·61-s + 1.17·63-s − 8.07·67-s + 1.34·69-s + ⋯ |
| L(s) = 1 | + 0.239·3-s − 0.156·7-s − 0.942·9-s + 0.426·11-s − 1.06·13-s + 0.242·17-s + 0.229·19-s − 0.0374·21-s + 0.676·23-s − 0.464·27-s − 0.339·29-s − 0.105·31-s + 0.101·33-s + 1.78·37-s − 0.253·39-s + 1.10·41-s + 0.951·43-s − 1.16·47-s − 0.975·49-s + 0.0580·51-s + 0.525·53-s + 0.0548·57-s − 1.50·59-s + 0.0750·61-s + 0.147·63-s − 0.986·67-s + 0.161·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 7 | \( 1 + 0.414T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 + 0.585T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 0.585T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 8.17T + 73T^{2} \) |
| 79 | \( 1 + 4.82T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.67231863420945891258511979446, −6.85845465646839794967814182019, −6.11505944747201651727072693718, −5.46872027428920782492720767828, −4.70522009819694583292009766514, −3.90579103523269888896598860152, −2.94072556248092215428615500613, −2.49838564805811716779720436384, −1.25348210809507190295900154157, 0,
1.25348210809507190295900154157, 2.49838564805811716779720436384, 2.94072556248092215428615500613, 3.90579103523269888896598860152, 4.70522009819694583292009766514, 5.46872027428920782492720767828, 6.11505944747201651727072693718, 6.85845465646839794967814182019, 7.67231863420945891258511979446
