| L(s) = 1 | + 3-s + 7-s − 2·9-s + 2·11-s + 13-s + 8·17-s − 2·19-s + 21-s + 23-s − 5·27-s + 29-s + 7·31-s + 2·33-s − 10·37-s + 39-s − 5·41-s − 8·43-s − 3·47-s + 49-s + 8·51-s + 10·53-s − 2·57-s + 8·59-s − 2·61-s − 2·63-s − 2·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s + 1.94·17-s − 0.458·19-s + 0.218·21-s + 0.208·23-s − 0.962·27-s + 0.185·29-s + 1.25·31-s + 0.348·33-s − 1.64·37-s + 0.160·39-s − 0.780·41-s − 1.21·43-s − 0.437·47-s + 1/7·49-s + 1.12·51-s + 1.37·53-s − 0.264·57-s + 1.04·59-s − 0.256·61-s − 0.251·63-s − 0.244·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.344034686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.344034686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.30498353918590, −13.68838384907074, −13.53172300177792, −12.68576353710592, −12.09629110365057, −11.75278183371081, −11.39647449496917, −10.54746679781824, −10.07563790090670, −9.790417780977810, −8.838146920912507, −8.572347430980223, −8.270268626152398, −7.478819761501975, −7.111703091608160, −6.262715268599133, −5.889727822921471, −5.154122435572158, −4.749018351818135, −3.754683974071105, −3.443445309116961, −2.875562486954322, −2.028969135778144, −1.415679071666514, −0.6103585345853220,
0.6103585345853220, 1.415679071666514, 2.028969135778144, 2.875562486954322, 3.443445309116961, 3.754683974071105, 4.749018351818135, 5.154122435572158, 5.889727822921471, 6.262715268599133, 7.111703091608160, 7.478819761501975, 8.270268626152398, 8.572347430980223, 8.838146920912507, 9.790417780977810, 10.07563790090670, 10.54746679781824, 11.39647449496917, 11.75278183371081, 12.09629110365057, 12.68576353710592, 13.53172300177792, 13.68838384907074, 14.30498353918590
