| L(s) = 1 | + 2·5-s − 7-s + 2·11-s + 2·13-s − 2·17-s + 19-s − 6·23-s − 25-s + 8·31-s − 2·35-s + 6·37-s + 8·43-s + 6·47-s + 49-s + 4·55-s + 10·61-s + 4·65-s + 4·67-s − 8·71-s + 6·73-s − 2·77-s − 8·79-s − 6·83-s − 4·85-s + 8·89-s − 2·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.603·11-s + 0.554·13-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 1/5·25-s + 1.43·31-s − 0.338·35-s + 0.986·37-s + 1.21·43-s + 0.875·47-s + 1/7·49-s + 0.539·55-s + 1.28·61-s + 0.496·65-s + 0.488·67-s − 0.949·71-s + 0.702·73-s − 0.227·77-s − 0.900·79-s − 0.658·83-s − 0.433·85-s + 0.847·89-s − 0.209·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.358631092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.358631092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.369957576806470730042697469395, −7.55824816512863986169112032651, −6.63950471091123955223945426540, −6.09827076954569557762365258618, −5.61382453770497446170760864869, −4.45380207567394009332607936005, −3.85108594955125640162344451216, −2.74500890226059552917810012261, −1.96786711767249338800192239096, −0.866090107514273170957529972213,
0.866090107514273170957529972213, 1.96786711767249338800192239096, 2.74500890226059552917810012261, 3.85108594955125640162344451216, 4.45380207567394009332607936005, 5.61382453770497446170760864869, 6.09827076954569557762365258618, 6.63950471091123955223945426540, 7.55824816512863986169112032651, 8.369957576806470730042697469395
