| L(s) = 1 | − 3-s − 7-s + 9-s + 3·11-s + 13-s + 7·17-s + 21-s + 7·23-s − 27-s − 4·29-s + 8·31-s − 3·33-s + 5·37-s − 39-s − 3·41-s + 8·43-s − 6·47-s − 6·49-s − 7·51-s + 11·53-s − 4·59-s + 61-s − 63-s − 12·67-s − 7·69-s − 9·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 1.69·17-s + 0.218·21-s + 1.45·23-s − 0.192·27-s − 0.742·29-s + 1.43·31-s − 0.522·33-s + 0.821·37-s − 0.160·39-s − 0.468·41-s + 1.21·43-s − 0.875·47-s − 6/7·49-s − 0.980·51-s + 1.51·53-s − 0.520·59-s + 0.128·61-s − 0.125·63-s − 1.46·67-s − 0.842·69-s − 1.06·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.959145417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.959145417\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−7.62494262271871471811213651890, −7.22126377339030391471896029577, −6.27406823965951843810127734362, −5.95675226497353076150822307162, −5.07792878991517465701598807997, −4.36622394107016148430873999216, −3.49128350408440612000270717550, −2.86202430318556327889806168318, −1.48050800838502808648840790286, −0.795922767386550153827753778796,
0.795922767386550153827753778796, 1.48050800838502808648840790286, 2.86202430318556327889806168318, 3.49128350408440612000270717550, 4.36622394107016148430873999216, 5.07792878991517465701598807997, 5.95675226497353076150822307162, 6.27406823965951843810127734362, 7.22126377339030391471896029577, 7.62494262271871471811213651890
