| L(s) = 1 | − 2·11-s − 13-s + 2·17-s + 2·19-s + 2·23-s + 6·29-s + 2·31-s + 6·37-s − 2·41-s − 6·43-s − 8·47-s − 7·49-s − 2·53-s − 6·59-s − 14·61-s − 10·71-s + 2·73-s − 4·79-s + 12·83-s + 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.277·13-s + 0.485·17-s + 0.458·19-s + 0.417·23-s + 1.11·29-s + 0.359·31-s + 0.986·37-s − 0.312·41-s − 0.914·43-s − 1.16·47-s − 49-s − 0.274·53-s − 0.781·59-s − 1.79·61-s − 1.18·71-s + 0.234·73-s − 0.450·79-s + 1.31·83-s + 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.83552341537276, −15.04510117737649, −14.78817937168717, −14.08619801511391, −13.51346335063836, −13.13295231263030, −12.39028879273038, −12.02182478025618, −11.36001547529298, −10.83851413293360, −10.11090797159539, −9.838213936361940, −9.101451396240105, −8.486436655695726, −7.761816903874358, −7.568205907167182, −6.520576301887896, −6.269546716800887, −5.282819787219858, −4.894455252497618, −4.249926001893433, −3.119540267411355, −2.988475181982010, −1.889672816169059, −1.080280076097553, 0,
1.080280076097553, 1.889672816169059, 2.988475181982010, 3.119540267411355, 4.249926001893433, 4.894455252497618, 5.282819787219858, 6.269546716800887, 6.520576301887896, 7.568205907167182, 7.761816903874358, 8.486436655695726, 9.101451396240105, 9.838213936361940, 10.11090797159539, 10.83851413293360, 11.36001547529298, 12.02182478025618, 12.39028879273038, 13.13295231263030, 13.51346335063836, 14.08619801511391, 14.78817937168717, 15.04510117737649, 15.83552341537276
