| L(s) = 1 | + 1.41·3-s − 0.999·9-s + 2·11-s + 2.82·13-s + 4.24·17-s − 4.24·19-s + 8·23-s − 5·25-s − 5.65·27-s + 6·29-s + 8.48·31-s + 2.82·33-s − 2·37-s + 4.00·39-s − 4.24·41-s + 6·43-s − 2.82·47-s + 6·51-s + 6·53-s − 6·57-s + 12.7·59-s − 5.65·61-s + 12·67-s + 11.3·69-s − 4·71-s − 1.41·73-s − 7.07·75-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.333·9-s + 0.603·11-s + 0.784·13-s + 1.02·17-s − 0.973·19-s + 1.66·23-s − 25-s − 1.08·27-s + 1.11·29-s + 1.52·31-s + 0.492·33-s − 0.328·37-s + 0.640·39-s − 0.662·41-s + 0.914·43-s − 0.412·47-s + 0.840·51-s + 0.824·53-s − 0.794·57-s + 1.65·59-s − 0.724·61-s + 1.46·67-s + 1.36·69-s − 0.474·71-s − 0.165·73-s − 0.816·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.383476848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.383476848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−9.316136142031300408698804928428, −8.473114568179355300384736496876, −8.194046697109229900159273724512, −7.03152444091693357187156538595, −6.26071282459919080446857332955, −5.34178652615149445904380531671, −4.18197883597665849303106895228, −3.35551205628358491016998314902, −2.48258315435609128359806847864, −1.12552427816405528035864022651,
1.12552427816405528035864022651, 2.48258315435609128359806847864, 3.35551205628358491016998314902, 4.18197883597665849303106895228, 5.34178652615149445904380531671, 6.26071282459919080446857332955, 7.03152444091693357187156538595, 8.194046697109229900159273724512, 8.473114568179355300384736496876, 9.316136142031300408698804928428
