| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 4·11-s + 12-s + 3·13-s + 14-s + 16-s − 5·17-s − 18-s − 21-s − 4·22-s − 23-s − 24-s − 5·25-s − 3·26-s + 27-s − 28-s + 10·29-s − 4·31-s − 32-s + 4·33-s + 5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.218·21-s − 0.852·22-s − 0.208·23-s − 0.204·24-s − 25-s − 0.588·26-s + 0.192·27-s − 0.188·28-s + 1.85·29-s − 0.718·31-s − 0.176·32-s + 0.696·33-s + 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.77624206496587, −12.16317127795048, −11.90099763403714, −11.32440543666459, −10.95082499497129, −10.45566502574032, −9.942883823383559, −9.440080850671487, −9.205094554632452, −8.610282144585102, −8.395549785520498, −7.887288359603973, −7.107176443469245, −6.951399508546149, −6.324946814019018, −6.021342741530424, −5.455117314368356, −4.446745701138802, −4.223132165863063, −3.699634228559502, −3.137725590023859, −2.394623216133884, −2.151447797675820, −1.244947706578543, −0.9320203284401695, 0,
0.9320203284401695, 1.244947706578543, 2.151447797675820, 2.394623216133884, 3.137725590023859, 3.699634228559502, 4.223132165863063, 4.446745701138802, 5.455117314368356, 6.021342741530424, 6.324946814019018, 6.951399508546149, 7.107176443469245, 7.887288359603973, 8.395549785520498, 8.610282144585102, 9.205094554632452, 9.440080850671487, 9.942883823383559, 10.45566502574032, 10.95082499497129, 11.32440543666459, 11.90099763403714, 12.16317127795048, 12.77624206496587
