| L(s) = 1 | + 2·3-s − 5-s + 9-s − 5·11-s − 6·13-s − 2·15-s − 7·19-s − 2·23-s + 25-s − 4·27-s + 9·29-s + 8·31-s − 10·33-s − 12·39-s + 7·41-s − 10·43-s − 45-s + 8·47-s − 4·53-s + 5·55-s − 14·57-s − 9·59-s − 5·61-s + 6·65-s − 4·67-s − 4·69-s − 71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-s − 1.66·13-s − 0.516·15-s − 1.60·19-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.67·29-s + 1.43·31-s − 1.74·33-s − 1.92·39-s + 1.09·41-s − 1.52·43-s − 0.149·45-s + 1.16·47-s − 0.549·53-s + 0.674·55-s − 1.85·57-s − 1.17·59-s − 0.640·61-s + 0.744·65-s − 0.488·67-s − 0.481·69-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.19925760653327, −12.72513518303012, −12.36656774530541, −11.97275468239185, −11.40953661790287, −10.70245751934324, −10.38110904236689, −9.991418847782116, −9.565204250276867, −8.863146981509607, −8.455548627573433, −8.182865262353786, −7.607387141056783, −7.454663631097864, −6.691800910137894, −6.206401544106840, −5.589751174590955, −4.874239027954514, −4.514079920539483, −4.213627829817771, −3.224268177587470, −2.904170423028967, −2.409778240928761, −2.173911706959361, −1.182842534472995, 0, 0,
1.182842534472995, 2.173911706959361, 2.409778240928761, 2.904170423028967, 3.224268177587470, 4.213627829817771, 4.514079920539483, 4.874239027954514, 5.589751174590955, 6.206401544106840, 6.691800910137894, 7.454663631097864, 7.607387141056783, 8.182865262353786, 8.455548627573433, 8.863146981509607, 9.565204250276867, 9.991418847782116, 10.38110904236689, 10.70245751934324, 11.40953661790287, 11.97275468239185, 12.36656774530541, 12.72513518303012, 13.19925760653327
