| L(s) = 1 | − 2·4-s − 5-s − 3·7-s − 3·11-s + 4·16-s + 3·17-s + 2·20-s − 3·23-s + 25-s + 6·28-s + 6·29-s − 6·31-s + 3·35-s + 9·37-s − 3·41-s + 10·43-s + 6·44-s − 12·47-s + 2·49-s + 3·53-s + 3·55-s + 12·59-s + 61-s − 8·64-s − 6·68-s + 9·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 1.13·7-s − 0.904·11-s + 16-s + 0.727·17-s + 0.447·20-s − 0.625·23-s + 1/5·25-s + 1.13·28-s + 1.11·29-s − 1.07·31-s + 0.507·35-s + 1.47·37-s − 0.468·41-s + 1.52·43-s + 0.904·44-s − 1.75·47-s + 2/7·49-s + 0.412·53-s + 0.404·55-s + 1.56·59-s + 0.128·61-s − 64-s − 0.727·68-s + 1.06·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−7.80344389015031456726171847930, −6.83498473250696354858237192721, −6.05884994788764195618764027139, −5.39743032287475146835085520396, −4.68580456220983475332227237791, −3.84470487948133896657873858784, −3.30050290355399583786502054577, −2.44655955775414105234625405529, −0.914729686813993554613629523337, 0,
0.914729686813993554613629523337, 2.44655955775414105234625405529, 3.30050290355399583786502054577, 3.84470487948133896657873858784, 4.68580456220983475332227237791, 5.39743032287475146835085520396, 6.05884994788764195618764027139, 6.83498473250696354858237192721, 7.80344389015031456726171847930
