| L(s) = 1 | − 3-s + 4·5-s + 9-s − 6·11-s − 5·13-s − 4·15-s − 2·17-s + 19-s + 6·23-s + 11·25-s − 27-s − 3·31-s + 6·33-s + 3·37-s + 5·39-s + 6·41-s − 5·43-s + 4·45-s − 4·47-s + 2·51-s − 6·53-s − 24·55-s − 57-s − 6·59-s + 2·61-s − 20·65-s − 7·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.80·11-s − 1.38·13-s − 1.03·15-s − 0.485·17-s + 0.229·19-s + 1.25·23-s + 11/5·25-s − 0.192·27-s − 0.538·31-s + 1.04·33-s + 0.493·37-s + 0.800·39-s + 0.937·41-s − 0.762·43-s + 0.596·45-s − 0.583·47-s + 0.280·51-s − 0.824·53-s − 3.23·55-s − 0.132·57-s − 0.781·59-s + 0.256·61-s − 2.48·65-s − 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.76083121635499773922310150746, −7.15629603812464777872984902693, −6.37138188120389108681228508431, −5.61542015526761598110175686073, −5.11171678577489223696591930622, −4.65504726285428168174856971959, −2.90014845106936895403041373653, −2.47572121547281430768161755633, −1.48079557801675043244696467133, 0,
1.48079557801675043244696467133, 2.47572121547281430768161755633, 2.90014845106936895403041373653, 4.65504726285428168174856971959, 5.11171678577489223696591930622, 5.61542015526761598110175686073, 6.37138188120389108681228508431, 7.15629603812464777872984902693, 7.76083121635499773922310150746
