| L(s) = 1 | − 2.56·3-s + 0.561·5-s − 7-s + 3.56·9-s + 11-s + 3.12·13-s − 1.43·15-s − 2·17-s − 5.12·19-s + 2.56·21-s − 1.43·23-s − 4.68·25-s − 1.43·27-s − 2·29-s − 10.5·31-s − 2.56·33-s − 0.561·35-s + 4.56·37-s − 8·39-s − 10·41-s + 4·43-s + 2.00·45-s − 6.24·47-s + 49-s + 5.12·51-s − 4.24·53-s + 0.561·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.251·5-s − 0.377·7-s + 1.18·9-s + 0.301·11-s + 0.866·13-s − 0.371·15-s − 0.485·17-s − 1.17·19-s + 0.558·21-s − 0.299·23-s − 0.936·25-s − 0.276·27-s − 0.371·29-s − 1.89·31-s − 0.445·33-s − 0.0949·35-s + 0.749·37-s − 1.28·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s − 0.911·47-s + 0.142·49-s + 0.717·51-s − 0.583·53-s + 0.0757·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 0.561T + 5T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 2.56T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 8.56T + 97T^{2} \) |
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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
−10.43913578322948856267191085054, −9.484333193830727732894361336105, −8.527295296177033558571543689271, −7.23420393626824865651930951731, −6.24770212755611380422417529504, −5.89832044716397750489797354287, −4.72563752578301480791720225628, −3.67537811329081465669048290880, −1.77691171676713607451976111874, 0,
1.77691171676713607451976111874, 3.67537811329081465669048290880, 4.72563752578301480791720225628, 5.89832044716397750489797354287, 6.24770212755611380422417529504, 7.23420393626824865651930951731, 8.527295296177033558571543689271, 9.484333193830727732894361336105, 10.43913578322948856267191085054
