| L(s) = 1 | + 0.414·2-s + 2.41·3-s − 1.82·4-s + 0.999·6-s − 1.58·8-s + 2.82·9-s + 4.82·11-s − 4.41·12-s + 0.828·13-s + 3·16-s − 0.828·17-s + 1.17·18-s + 2.82·19-s + 1.99·22-s + 2.41·23-s − 3.82·24-s + 0.343·26-s − 0.414·27-s − 29-s + 6·31-s + 4.41·32-s + 11.6·33-s − 0.343·34-s − 5.17·36-s + 1.17·38-s + 1.99·39-s + 2.17·41-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + 1.39·3-s − 0.914·4-s + 0.408·6-s − 0.560·8-s + 0.942·9-s + 1.45·11-s − 1.27·12-s + 0.229·13-s + 0.750·16-s − 0.200·17-s + 0.276·18-s + 0.648·19-s + 0.426·22-s + 0.503·23-s − 0.781·24-s + 0.0672·26-s − 0.0797·27-s − 0.185·29-s + 1.07·31-s + 0.780·32-s + 2.02·33-s − 0.0588·34-s − 0.861·36-s + 0.190·38-s + 0.320·39-s + 0.339·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.684676762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.684676762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−9.508420106994809935996078002804, −8.808610552199473868203134978649, −8.444495872034550160721314262673, −7.41456512257279403945557670395, −6.45648019926991244213253100313, −5.34011782243269297055595264148, −4.20784196868123022900594135744, −3.66424507225266072392019886067, −2.71726020941408614209350142015, −1.23607802281236117297335351951,
1.23607802281236117297335351951, 2.71726020941408614209350142015, 3.66424507225266072392019886067, 4.20784196868123022900594135744, 5.34011782243269297055595264148, 6.45648019926991244213253100313, 7.41456512257279403945557670395, 8.444495872034550160721314262673, 8.808610552199473868203134978649, 9.508420106994809935996078002804
