| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3·7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 2·13-s − 3·14-s − 15-s + 16-s + 2·17-s − 18-s − 20-s + 3·21-s − 22-s + 23-s − 24-s + 25-s + 2·26-s + 27-s + 3·28-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.654·21-s − 0.213·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.566·28-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 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
−16.86127719662636, −16.33444324940232, −15.56346982263631, −15.15959844300378, −14.47807114219386, −14.29476656201405, −13.43018153986808, −12.69870080675718, −11.96582430323238, −11.68166055006001, −10.90850530752034, −10.38520567032846, −9.739865677043213, −9.007519216932497, −8.538751178049366, −7.980006881390608, −7.391443016731359, −6.967366255832157, −5.990184434818289, −5.086984331504731, −4.560524706642520, −3.585413461396830, −2.954261690715228, −1.870048763815578, −1.387156666629877, 0,
1.387156666629877, 1.870048763815578, 2.954261690715228, 3.585413461396830, 4.560524706642520, 5.086984331504731, 5.990184434818289, 6.967366255832157, 7.391443016731359, 7.980006881390608, 8.538751178049366, 9.007519216932497, 9.739865677043213, 10.38520567032846, 10.90850530752034, 11.68166055006001, 11.96582430323238, 12.69870080675718, 13.43018153986808, 14.29476656201405, 14.47807114219386, 15.15959844300378, 15.56346982263631, 16.33444324940232, 16.86127719662636
