| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 3·11-s − 2·13-s − 15-s − 3·17-s + 2·19-s − 21-s + 25-s − 27-s + 3·29-s − 31-s + 3·33-s + 35-s + 2·39-s + 9·41-s + 11·43-s + 45-s − 6·49-s + 3·51-s − 9·53-s − 3·55-s − 2·57-s − 6·59-s + 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s − 0.258·15-s − 0.727·17-s + 0.458·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.557·29-s − 0.179·31-s + 0.522·33-s + 0.169·35-s + 0.320·39-s + 1.40·41-s + 1.67·43-s + 0.149·45-s − 6/7·49-s + 0.420·51-s − 1.23·53-s − 0.404·55-s − 0.264·57-s − 0.781·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.83268026276553, −12.45404098143408, −11.92865987196097, −11.36565677549204, −10.99892047673804, −10.63375801360049, −10.20074561365041, −9.636974579813270, −9.238757775915722, −8.846450406390426, −8.078557721844762, −7.643469104317919, −7.459820828783597, −6.636343870288568, −6.299208015742298, −5.806276299806115, −5.237843325963980, −4.845763249878544, −4.498996265916447, −3.816513758002545, −3.095526776910180, −2.490261240012067, −2.148286883438661, −1.368650263640862, −0.7323239368784788, 0,
0.7323239368784788, 1.368650263640862, 2.148286883438661, 2.490261240012067, 3.095526776910180, 3.816513758002545, 4.498996265916447, 4.845763249878544, 5.237843325963980, 5.806276299806115, 6.299208015742298, 6.636343870288568, 7.459820828783597, 7.643469104317919, 8.078557721844762, 8.846450406390426, 9.238757775915722, 9.636974579813270, 10.20074561365041, 10.63375801360049, 10.99892047673804, 11.36565677549204, 11.92865987196097, 12.45404098143408, 12.83268026276553
