L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s + 2·11-s + 13-s + 2·15-s + 6·17-s − 2·19-s − 2·21-s − 25-s − 27-s − 2·29-s − 2·31-s − 2·33-s − 4·35-s − 10·37-s − 39-s + 2·41-s + 8·43-s − 2·45-s + 2·47-s − 3·49-s − 6·51-s + 2·53-s − 4·55-s + 2·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.516·15-s + 1.45·17-s − 0.458·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.348·33-s − 0.676·35-s − 1.64·37-s − 0.160·39-s + 0.312·41-s + 1.21·43-s − 0.298·45-s + 0.291·47-s − 3/7·49-s − 0.840·51-s + 0.274·53-s − 0.539·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.379837304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.379837304\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.818453540494309832076182894520, −8.022301336260565833613749901805, −7.49513042391929280330744368515, −6.66074903907656898060316231781, −5.70848705946811899102526270575, −5.03511394063434846108846235628, −4.04128971437181216870767918774, −3.49664739136305444125445933691, −1.93035310785089381364797871826, −0.794015071717988881268874676005,
0.794015071717988881268874676005, 1.93035310785089381364797871826, 3.49664739136305444125445933691, 4.04128971437181216870767918774, 5.03511394063434846108846235628, 5.70848705946811899102526270575, 6.66074903907656898060316231781, 7.49513042391929280330744368515, 8.022301336260565833613749901805, 8.818453540494309832076182894520