| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 0.449·11-s − 0.449·13-s − 2·14-s + 16-s − 4.89·17-s + 4.89·19-s − 20-s + 0.449·22-s − 2·23-s + 25-s − 0.449·26-s − 2·28-s − 10.4·29-s − 31-s + 32-s − 4.89·34-s + 2·35-s + 4.44·37-s + 4.89·38-s − 40-s − 5.55·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.135·11-s − 0.124·13-s − 0.534·14-s + 0.250·16-s − 1.18·17-s + 1.12·19-s − 0.223·20-s + 0.0958·22-s − 0.417·23-s + 0.200·25-s − 0.0881·26-s − 0.377·28-s − 1.94·29-s − 0.179·31-s + 0.176·32-s − 0.840·34-s + 0.338·35-s + 0.731·37-s + 0.794·38-s − 0.158·40-s − 0.846·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 0.449T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5.55T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 5.34T + 53T^{2} \) |
| 59 | \( 1 - 0.898T + 59T^{2} \) |
| 61 | \( 1 + 6.44T + 61T^{2} \) |
| 67 | \( 1 + 3.10T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 0.898T + 97T^{2} \) |
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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
−8.308458983552151283740693531222, −7.49403275224117371582034366162, −6.84795676934113687581015258470, −6.11122551504348560775803132544, −5.29397591488532094900707504043, −4.39821472824005136079120561370, −3.62171219375031205202388843566, −2.88748495525543559393267780226, −1.71577376471700504147179695759, 0,
1.71577376471700504147179695759, 2.88748495525543559393267780226, 3.62171219375031205202388843566, 4.39821472824005136079120561370, 5.29397591488532094900707504043, 6.11122551504348560775803132544, 6.84795676934113687581015258470, 7.49403275224117371582034366162, 8.308458983552151283740693531222
