| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 0.133·7-s + 8-s + 9-s + 2.92·11-s − 12-s + 4.92·13-s − 0.133·14-s + 16-s + 3.13·17-s + 18-s − 0.266·19-s + 0.133·21-s + 2.92·22-s + 2·23-s − 24-s + 4.92·26-s − 27-s − 0.133·28-s + 3.13·29-s − 31-s + 32-s − 2.92·33-s + 3.13·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.0504·7-s + 0.353·8-s + 0.333·9-s + 0.881·11-s − 0.288·12-s + 1.36·13-s − 0.0356·14-s + 0.250·16-s + 0.759·17-s + 0.235·18-s − 0.0612·19-s + 0.0291·21-s + 0.623·22-s + 0.417·23-s − 0.204·24-s + 0.965·26-s − 0.192·27-s − 0.0252·28-s + 0.581·29-s − 0.179·31-s + 0.176·32-s − 0.509·33-s + 0.537·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.056400991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.056400991\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 0.133T + 7T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 - 4.92T + 13T^{2} \) |
| 17 | \( 1 - 3.13T + 17T^{2} \) |
| 19 | \( 1 + 0.266T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 3.13T + 29T^{2} \) |
| 37 | \( 1 - 0.924T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + 6.84T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 9.84T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 8.71T + 97T^{2} \) |
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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
−8.343758761426842555047786304184, −7.33752787064830191040773383989, −6.60832806854210189179211412793, −6.15444031687452024189322047399, −5.37521498960997554902038239874, −4.65489861837693091810961611592, −3.72958304866677299021993123494, −3.24861434774606724455858577579, −1.82721525927059732864633649858, −0.967172360582310006964082412142,
0.967172360582310006964082412142, 1.82721525927059732864633649858, 3.24861434774606724455858577579, 3.72958304866677299021993123494, 4.65489861837693091810961611592, 5.37521498960997554902038239874, 6.15444031687452024189322047399, 6.60832806854210189179211412793, 7.33752787064830191040773383989, 8.343758761426842555047786304184
