| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 4·11-s − 2·13-s + 16-s − 6·17-s − 4·19-s + 20-s − 4·22-s + 25-s + 2·26-s + 2·29-s − 32-s + 6·34-s + 2·37-s + 4·38-s − 40-s − 10·41-s + 4·43-s + 4·44-s − 7·49-s − 50-s − 2·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 1/5·25-s + 0.392·26-s + 0.371·29-s − 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 1.56·41-s + 0.609·43-s + 0.603·44-s − 49-s − 0.141·50-s − 0.277·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.465784971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.465784971\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−14.63583834531186, −14.15529926578204, −13.56569864261810, −12.96817577520123, −12.54203554153888, −11.83291679781334, −11.46551352224261, −10.90647615089492, −10.37203085799669, −9.807925220293249, −9.326946060050148, −8.865941054746976, −8.388479473413364, −7.833757117101843, −6.917343604038695, −6.591944762282501, −6.370712201223495, −5.365957257682898, −4.847383938989632, −4.052922188366528, −3.575641432250237, −2.432577965122565, −2.202579245938331, −1.359708174199796, −0.4883917756784333,
0.4883917756784333, 1.359708174199796, 2.202579245938331, 2.432577965122565, 3.575641432250237, 4.052922188366528, 4.847383938989632, 5.365957257682898, 6.370712201223495, 6.591944762282501, 6.917343604038695, 7.833757117101843, 8.388479473413364, 8.865941054746976, 9.326946060050148, 9.807925220293249, 10.37203085799669, 10.90647615089492, 11.46551352224261, 11.83291679781334, 12.54203554153888, 12.96817577520123, 13.56569864261810, 14.15529926578204, 14.63583834531186
