| L(s) = 1 | − 3·3-s + 6·9-s − 5·11-s − 5·13-s − 7·17-s + 2·19-s + 2·23-s − 9·27-s + 7·29-s − 4·31-s + 15·33-s + 6·37-s + 15·39-s + 12·41-s + 2·43-s + 47-s + 21·51-s − 6·57-s + 4·59-s − 4·61-s − 8·67-s − 6·69-s + 6·73-s − 3·79-s + 9·81-s − 4·83-s − 21·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 1.50·11-s − 1.38·13-s − 1.69·17-s + 0.458·19-s + 0.417·23-s − 1.73·27-s + 1.29·29-s − 0.718·31-s + 2.61·33-s + 0.986·37-s + 2.40·39-s + 1.87·41-s + 0.304·43-s + 0.145·47-s + 2.94·51-s − 0.794·57-s + 0.520·59-s − 0.512·61-s − 0.977·67-s − 0.722·69-s + 0.702·73-s − 0.337·79-s + 81-s − 0.439·83-s − 2.25·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.40725872385309314379193622967, −6.50768200864736685126772093855, −5.95333953952226155933565448632, −5.19942253634729679031764081940, −4.76608856833930622168121606062, −4.26572122925126721012026731763, −2.80835496541964668704423108192, −2.19379526947480064353524210804, −0.801943016196788934288746029848, 0,
0.801943016196788934288746029848, 2.19379526947480064353524210804, 2.80835496541964668704423108192, 4.26572122925126721012026731763, 4.76608856833930622168121606062, 5.19942253634729679031764081940, 5.95333953952226155933565448632, 6.50768200864736685126772093855, 7.40725872385309314379193622967
