| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s − 2·9-s + 3·11-s + 12-s + 13-s − 14-s + 16-s + 3·17-s − 2·18-s + 6·19-s − 21-s + 3·22-s − 8·23-s + 24-s + 26-s − 5·27-s − 28-s − 6·29-s + 2·31-s + 32-s + 3·33-s + 3·34-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 − 2/3·9-s + 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 1.37·19-s − 0.218·21-s + 0.639·22-s − 1.66·23-s + 0.204·24-s + 0.196·26-s − 0.962·27-s − 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.546575073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.546575073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 293 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.89919015775855, −13.35866092357523, −12.89081879054501, −12.09957973926822, −11.86244793847310, −11.57714130479518, −10.88300942072167, −10.21330709150827, −9.791068898298840, −9.244074090740423, −8.871698060354251, −8.035975713788220, −7.827647819572627, −7.212991389902249, −6.459465885072371, −6.178165216809863, −5.414998999766041, −5.244254369424946, −4.220484361998352, −3.670328970266684, −3.435803784582618, −2.808917712784620, −2.030600518204334, −1.508536824351960, −0.5507361657334389,
0.5507361657334389, 1.508536824351960, 2.030600518204334, 2.808917712784620, 3.435803784582618, 3.670328970266684, 4.220484361998352, 5.244254369424946, 5.414998999766041, 6.178165216809863, 6.459465885072371, 7.212991389902249, 7.827647819572627, 8.035975713788220, 8.871698060354251, 9.244074090740423, 9.791068898298840, 10.21330709150827, 10.88300942072167, 11.57714130479518, 11.86244793847310, 12.09957973926822, 12.89081879054501, 13.35866092357523, 13.89919015775855
