| L(s) = 1 | + 3-s − 5·7-s + 9-s − 6·11-s − 3·13-s − 2·17-s + 19-s − 5·21-s − 2·23-s + 27-s + 6·29-s + 3·31-s − 6·33-s − 6·37-s − 3·39-s + 4·41-s + 11·43-s − 10·47-s + 18·49-s − 2·51-s − 8·53-s + 57-s − 6·59-s + 3·61-s − 5·63-s − 67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s − 1.80·11-s − 0.832·13-s − 0.485·17-s + 0.229·19-s − 1.09·21-s − 0.417·23-s + 0.192·27-s + 1.11·29-s + 0.538·31-s − 1.04·33-s − 0.986·37-s − 0.480·39-s + 0.624·41-s + 1.67·43-s − 1.45·47-s + 18/7·49-s − 0.280·51-s − 1.09·53-s + 0.132·57-s − 0.781·59-s + 0.384·61-s − 0.629·63-s − 0.122·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−10.01640260119628208124792470973, −9.573634131463927078112432516263, −8.483378982015133633479563111090, −7.55110806087691558837157186820, −6.72557170085953961720882600963, −5.70500717305478856615746032890, −4.50519787498977221160424199992, −3.11187956794533611218111257703, −2.55892028284702124429993107097, 0,
2.55892028284702124429993107097, 3.11187956794533611218111257703, 4.50519787498977221160424199992, 5.70500717305478856615746032890, 6.72557170085953961720882600963, 7.55110806087691558837157186820, 8.483378982015133633479563111090, 9.573634131463927078112432516263, 10.01640260119628208124792470973
