| L(s) = 1 | − 2-s + 0.655·3-s + 4-s + 2.17·5-s − 0.655·6-s − 8-s − 2.57·9-s − 2.17·10-s + 0.182·11-s + 0.655·12-s − 1.49·13-s + 1.42·15-s + 16-s − 4.61·17-s + 2.57·18-s − 0.344·19-s + 2.17·20-s − 0.182·22-s − 1.55·23-s − 0.655·24-s − 0.279·25-s + 1.49·26-s − 3.64·27-s + 29-s − 1.42·30-s + 8.74·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.378·3-s + 0.5·4-s + 0.971·5-s − 0.267·6-s − 0.353·8-s − 0.856·9-s − 0.687·10-s + 0.0550·11-s + 0.189·12-s − 0.414·13-s + 0.367·15-s + 0.250·16-s − 1.12·17-s + 0.605·18-s − 0.0791·19-s + 0.485·20-s − 0.0389·22-s − 0.323·23-s − 0.133·24-s − 0.0558·25-s + 0.293·26-s − 0.702·27-s + 0.185·29-s − 0.259·30-s + 1.57·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.655T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 11 | \( 1 - 0.182T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 + 0.344T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 + 5.84T + 53T^{2} \) |
| 59 | \( 1 + 1.53T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 2.37T + 67T^{2} \) |
| 71 | \( 1 - 2.17T + 71T^{2} \) |
| 73 | \( 1 - 8.73T + 73T^{2} \) |
| 79 | \( 1 - 8.92T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 7.04T + 89T^{2} \) |
| 97 | \( 1 + 4.15T + 97T^{2} \) |
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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
−8.497905948193227281787631604880, −7.911534438062839068159022948145, −6.83409154245527559922193731304, −6.28111997588082548695294831055, −5.48195353958409307677702654985, −4.54250189184838750698613817463, −3.23039172578527183245737146903, −2.42643309435168845272111350240, −1.66142071476846615663474914407, 0,
1.66142071476846615663474914407, 2.42643309435168845272111350240, 3.23039172578527183245737146903, 4.54250189184838750698613817463, 5.48195353958409307677702654985, 6.28111997588082548695294831055, 6.83409154245527559922193731304, 7.911534438062839068159022948145, 8.497905948193227281787631604880
