L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 3·8-s + 9-s − 10-s − 4·11-s + 12-s − 2·13-s − 15-s − 16-s + 2·17-s − 18-s + 4·19-s − 20-s + 4·22-s − 3·24-s + 25-s + 2·26-s − 27-s − 2·29-s + 30-s − 5·32-s + 4·33-s − 2·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.883·32-s + 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3501507605\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3501507605\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−19.11046245900217536106846201128, −18.10585375705648027116926624524, −17.23595347746433122931066197428, −15.95902603015140726835865441199, −14.00692585049802430183598730465, −12.64617876135106218873747419153, −10.67892245123374144028858824682, −9.523451675812284265792380668213, −7.66488013441745380243230842523, −5.23920392624592057055772361749,
5.23920392624592057055772361749, 7.66488013441745380243230842523, 9.523451675812284265792380668213, 10.67892245123374144028858824682, 12.64617876135106218873747419153, 14.00692585049802430183598730465, 15.95902603015140726835865441199, 17.23595347746433122931066197428, 18.10585375705648027116926624524, 19.11046245900217536106846201128