| L(s) = 1 | − 2-s + 4-s − 1.64·5-s − 8-s + 1.64·10-s + 1.64·11-s + 0.645·13-s + 16-s + 1.64·17-s + 2·19-s − 1.64·20-s − 1.64·22-s − 9.29·23-s − 2.29·25-s − 0.645·26-s − 7.64·29-s + 0.645·31-s − 32-s − 1.64·34-s + 3.93·37-s − 2·38-s + 1.64·40-s + 4.93·41-s + 5·43-s + 1.64·44-s + 9.29·46-s + 10.9·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.736·5-s − 0.353·8-s + 0.520·10-s + 0.496·11-s + 0.179·13-s + 0.250·16-s + 0.399·17-s + 0.458·19-s − 0.368·20-s − 0.350·22-s − 1.93·23-s − 0.458·25-s − 0.126·26-s − 1.41·29-s + 0.115·31-s − 0.176·32-s − 0.282·34-s + 0.647·37-s − 0.324·38-s + 0.260·40-s + 0.771·41-s + 0.762·43-s + 0.248·44-s + 1.36·46-s + 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.64T + 5T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 0.645T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 9.29T + 23T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 - 0.645T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 7.58T + 97T^{2} \) |
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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
−8.399030497591828844802120377317, −7.70796465662326201273858237927, −7.31974379342419893642196146749, −6.16118333744205388885565266414, −5.64354273892781596066807266471, −4.22513355675809445930178184655, −3.71779790363866877236405743123, −2.50766523018405897724107243041, −1.36405521513631835469785042490, 0,
1.36405521513631835469785042490, 2.50766523018405897724107243041, 3.71779790363866877236405743123, 4.22513355675809445930178184655, 5.64354273892781596066807266471, 6.16118333744205388885565266414, 7.31974379342419893642196146749, 7.70796465662326201273858237927, 8.399030497591828844802120377317
