| L(s) = 1 | − 3-s − 5-s − 2·9-s − 5·11-s − 13-s + 15-s − 3·17-s − 19-s − 4·25-s + 5·27-s − 2·29-s + 5·33-s − 11·37-s + 39-s + 5·41-s − 11·43-s + 2·45-s + 4·47-s + 3·51-s + 5·53-s + 5·55-s + 57-s + 3·59-s − 6·61-s + 65-s + 3·67-s + 9·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s − 0.229·19-s − 4/5·25-s + 0.962·27-s − 0.371·29-s + 0.870·33-s − 1.80·37-s + 0.160·39-s + 0.780·41-s − 1.67·43-s + 0.298·45-s + 0.583·47-s + 0.420·51-s + 0.686·53-s + 0.674·55-s + 0.132·57-s + 0.390·59-s − 0.768·61-s + 0.124·65-s + 0.366·67-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.97120179340040, −12.31356951897597, −12.00157659255642, −11.67774526872698, −11.01324423356686, −10.70008385084342, −10.51254698342704, −9.797016403464364, −9.362654286581304, −8.756816899082726, −8.299289252470703, −7.960844579114035, −7.523881551433567, −6.756094634125610, −6.670573274947762, −5.843580430967515, −5.410490173066037, −5.130263609252767, −4.627410081089987, −3.927789969304053, −3.473016200037416, −2.836462071594634, −2.292785087917813, −1.877348981977221, −0.8480617358164774, 0, 0,
0.8480617358164774, 1.877348981977221, 2.292785087917813, 2.836462071594634, 3.473016200037416, 3.927789969304053, 4.627410081089987, 5.130263609252767, 5.410490173066037, 5.843580430967515, 6.670573274947762, 6.756094634125610, 7.523881551433567, 7.960844579114035, 8.299289252470703, 8.756816899082726, 9.362654286581304, 9.797016403464364, 10.51254698342704, 10.70008385084342, 11.01324423356686, 11.67774526872698, 12.00157659255642, 12.31356951897597, 12.97120179340040
