| L(s) = 1 | − 4·2-s − 3·3-s + 8·4-s − 13·5-s + 12·6-s − 26·7-s + 9·9-s + 52·10-s − 24·12-s + 73·13-s + 104·14-s + 39·15-s − 64·16-s − 31·17-s − 36·18-s + 108·19-s − 104·20-s + 78·21-s − 86·23-s + 44·25-s − 292·26-s − 27·27-s − 208·28-s + 207·29-s − 156·30-s + 208·31-s + 256·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 1.16·5-s + 0.816·6-s − 1.40·7-s + 1/3·9-s + 1.64·10-s − 0.577·12-s + 1.55·13-s + 1.98·14-s + 0.671·15-s − 16-s − 0.442·17-s − 0.471·18-s + 1.30·19-s − 1.16·20-s + 0.810·21-s − 0.779·23-s + 0.351·25-s − 2.20·26-s − 0.192·27-s − 1.40·28-s + 1.32·29-s − 0.949·30-s + 1.20·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 + 13 T + p^{3} T^{2} \) |
| 7 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 73 T + p^{3} T^{2} \) |
| 17 | \( 1 + 31 T + p^{3} T^{2} \) |
| 19 | \( 1 - 108 T + p^{3} T^{2} \) |
| 23 | \( 1 + 86 T + p^{3} T^{2} \) |
| 29 | \( 1 - 207 T + p^{3} T^{2} \) |
| 31 | \( 1 - 208 T + p^{3} T^{2} \) |
| 37 | \( 1 - 45 T + p^{3} T^{2} \) |
| 41 | \( 1 + 247 T + p^{3} T^{2} \) |
| 43 | \( 1 - 450 T + p^{3} T^{2} \) |
| 47 | \( 1 + 500 T + p^{3} T^{2} \) |
| 53 | \( 1 + 441 T + p^{3} T^{2} \) |
| 59 | \( 1 - 598 T + p^{3} T^{2} \) |
| 61 | \( 1 + 378 T + p^{3} T^{2} \) |
| 67 | \( 1 - 494 T + p^{3} T^{2} \) |
| 71 | \( 1 + 594 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1034 T + p^{3} T^{2} \) |
| 79 | \( 1 + 352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 360 T + p^{3} T^{2} \) |
| 89 | \( 1 + 351 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1079 T + p^{3} T^{2} \) |
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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
−10.35485827882074705004000985052, −9.690431455642004068773008326754, −8.659797398301779021238289511977, −7.907473805731824122793797774373, −6.87938942167902816935910704847, −6.12266902746513471057827191141, −4.35036333141957798385810531143, −3.19823779423267923983083649726, −1.04854441214814406397058049865, 0,
1.04854441214814406397058049865, 3.19823779423267923983083649726, 4.35036333141957798385810531143, 6.12266902746513471057827191141, 6.87938942167902816935910704847, 7.907473805731824122793797774373, 8.659797398301779021238289511977, 9.690431455642004068773008326754, 10.35485827882074705004000985052
