| L(s) = 1 | − 8·3-s − 5·5-s + 4·7-s + 37·9-s + 12·11-s + 58·13-s + 40·15-s + 66·17-s − 100·19-s − 32·21-s − 132·23-s + 25·25-s − 80·27-s + 90·29-s − 152·31-s − 96·33-s − 20·35-s + 34·37-s − 464·39-s − 438·41-s + 32·43-s − 185·45-s + 204·47-s − 327·49-s − 528·51-s − 222·53-s − 60·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.447·5-s + 0.215·7-s + 1.37·9-s + 0.328·11-s + 1.23·13-s + 0.688·15-s + 0.941·17-s − 1.20·19-s − 0.332·21-s − 1.19·23-s + 1/5·25-s − 0.570·27-s + 0.576·29-s − 0.880·31-s − 0.506·33-s − 0.0965·35-s + 0.151·37-s − 1.90·39-s − 1.66·41-s + 0.113·43-s − 0.612·45-s + 0.633·47-s − 0.953·49-s − 1.44·51-s − 0.575·53-s − 0.147·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 132 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 438 T + p^{3} T^{2} \) |
| 43 | \( 1 - 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 204 T + p^{3} T^{2} \) |
| 53 | \( 1 + 222 T + p^{3} T^{2} \) |
| 59 | \( 1 - 420 T + p^{3} T^{2} \) |
| 61 | \( 1 + 902 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 + 432 T + p^{3} T^{2} \) |
| 73 | \( 1 - 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 72 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1106 T + p^{3} T^{2} \) |
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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
−10.81973378223859344787114230451, −10.23761177653787897948257998029, −8.797984037868545978872730520829, −7.77035004080839675677941001595, −6.50091674456001770610585470102, −5.90618430189883161690154628463, −4.74858137452270198044189754233, −3.68736784694363347570436425678, −1.43021952791533749067371864952, 0,
1.43021952791533749067371864952, 3.68736784694363347570436425678, 4.74858137452270198044189754233, 5.90618430189883161690154628463, 6.50091674456001770610585470102, 7.77035004080839675677941001595, 8.797984037868545978872730520829, 10.23761177653787897948257998029, 10.81973378223859344787114230451
