| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 2·13-s + 14-s + 16-s − 2·17-s + 19-s + 20-s + 8·23-s + 25-s − 2·26-s − 28-s − 32-s + 2·34-s − 35-s − 6·37-s − 38-s − 40-s + 2·41-s − 10·43-s − 8·46-s − 8·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.229·19-s + 0.223·20-s + 1.66·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 0.986·37-s − 0.162·38-s − 0.158·40-s + 0.312·41-s − 1.52·43-s − 1.17·46-s − 1.16·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.74182698778999, −16.09499981566582, −15.75214994483889, −14.92058801730381, −14.64441773426837, −13.62970862516953, −13.30545019631362, −12.74347010484447, −11.96660216113770, −11.38864382640135, −10.78054204784130, −10.30554947886708, −9.629006088674872, −9.021486831077260, −8.680951537391092, −7.870436420736133, −7.141081162852767, −6.589947589685783, −6.063598212361945, −5.203480246760656, −4.577906234071380, −3.381914389928655, −2.992139085787969, −1.900865230704536, −1.201149699668572, 0,
1.201149699668572, 1.900865230704536, 2.992139085787969, 3.381914389928655, 4.577906234071380, 5.203480246760656, 6.063598212361945, 6.589947589685783, 7.141081162852767, 7.870436420736133, 8.680951537391092, 9.021486831077260, 9.629006088674872, 10.30554947886708, 10.78054204784130, 11.38864382640135, 11.96660216113770, 12.74347010484447, 13.30545019631362, 13.62970862516953, 14.64441773426837, 14.92058801730381, 15.75214994483889, 16.09499981566582, 16.74182698778999
