| L(s) = 1 | − 2·4-s + 5-s − 1.73·7-s + 4·16-s − 3.46·17-s + 5.19·19-s − 2·20-s − 6·23-s + 25-s + 3.46·28-s + 6.92·29-s + 31-s − 1.73·35-s − 5·37-s + 3.46·41-s − 10.3·43-s + 12·47-s − 4·49-s − 6·53-s + 12.1·61-s − 8·64-s − 5·67-s + 6.92·68-s + 6·71-s + 1.73·73-s − 10.3·76-s + 15.5·79-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.654·7-s + 16-s − 0.840·17-s + 1.19·19-s − 0.447·20-s − 1.25·23-s + 0.200·25-s + 0.654·28-s + 1.28·29-s + 0.179·31-s − 0.292·35-s − 0.821·37-s + 0.541·41-s − 1.58·43-s + 1.75·47-s − 0.571·49-s − 0.824·53-s + 1.55·61-s − 64-s − 0.610·67-s + 0.840·68-s + 0.712·71-s + 0.202·73-s − 1.19·76-s + 1.75·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−8.030411022038040963935798238524, −6.99250165850645537751584055318, −6.36424756758133174761541488458, −5.56132720129851857202182063052, −4.93858295195974718194769374186, −4.09624654899216395589804544139, −3.36617818308706827410639026401, −2.43500839787088700842339615987, −1.19494010893566666338565529033, 0,
1.19494010893566666338565529033, 2.43500839787088700842339615987, 3.36617818308706827410639026401, 4.09624654899216395589804544139, 4.93858295195974718194769374186, 5.56132720129851857202182063052, 6.36424756758133174761541488458, 6.99250165850645537751584055318, 8.030411022038040963935798238524
