| L(s) = 1 | + 3·3-s + 5·5-s − 7·7-s + 9·9-s − 42·11-s + 20·13-s + 15·15-s + 66·17-s − 38·19-s − 21·21-s − 12·23-s + 25·25-s + 27·27-s − 258·29-s − 146·31-s − 126·33-s − 35·35-s + 434·37-s + 60·39-s − 282·41-s − 20·43-s + 45·45-s + 72·47-s + 49·49-s + 198·51-s + 336·53-s − 210·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.15·11-s + 0.426·13-s + 0.258·15-s + 0.941·17-s − 0.458·19-s − 0.218·21-s − 0.108·23-s + 1/5·25-s + 0.192·27-s − 1.65·29-s − 0.845·31-s − 0.664·33-s − 0.169·35-s + 1.92·37-s + 0.246·39-s − 1.07·41-s − 0.0709·43-s + 0.149·45-s + 0.223·47-s + 1/7·49-s + 0.543·51-s + 0.870·53-s − 0.514·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + 42 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 258 T + p^{3} T^{2} \) |
| 31 | \( 1 + 146 T + p^{3} T^{2} \) |
| 37 | \( 1 - 434 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 20 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 336 T + p^{3} T^{2} \) |
| 59 | \( 1 - 360 T + p^{3} T^{2} \) |
| 61 | \( 1 + 682 T + p^{3} T^{2} \) |
| 67 | \( 1 + 812 T + p^{3} T^{2} \) |
| 71 | \( 1 + 810 T + p^{3} T^{2} \) |
| 73 | \( 1 + 124 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1136 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1208 T + p^{3} T^{2} \) |
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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
−8.638307577780672359234010356461, −7.77009943952594211233462479874, −7.21027022986347936511281085244, −6.00533045917114145715677395758, −5.48853096486770319047370649548, −4.31631132439366062534610889767, −3.33312142869497315328442703554, −2.51873453369019607688938975072, −1.47229320978480376234335810280, 0,
1.47229320978480376234335810280, 2.51873453369019607688938975072, 3.33312142869497315328442703554, 4.31631132439366062534610889767, 5.48853096486770319047370649548, 6.00533045917114145715677395758, 7.21027022986347936511281085244, 7.77009943952594211233462479874, 8.638307577780672359234010356461
