| L(s) = 1 | − 3-s − 2·4-s − 5-s − 3·7-s + 9-s − 3·11-s + 2·12-s − 13-s + 15-s + 4·16-s − 17-s + 6·19-s + 2·20-s + 3·21-s − 7·23-s + 25-s − 27-s + 6·28-s − 8·29-s + 31-s + 3·33-s + 3·35-s − 2·36-s − 9·37-s + 39-s − 5·41-s − 2·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s − 0.242·17-s + 1.37·19-s + 0.447·20-s + 0.654·21-s − 1.45·23-s + 1/5·25-s − 0.192·27-s + 1.13·28-s − 1.48·29-s + 0.179·31-s + 0.522·33-s + 0.507·35-s − 1/3·36-s − 1.47·37-s + 0.160·39-s − 0.780·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04778634009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04778634009\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.85142846098016081198835168386, −7.57825594242192797000800499426, −6.57058508708319604080006445584, −5.84126322781191730929599779377, −5.18216684769054257787794547017, −4.56440343433289668921138084062, −3.55177581344733015412600312264, −3.16080152176289073528751699268, −1.66790914896068438535915623091, −0.11677754306259318212730105999,
0.11677754306259318212730105999, 1.66790914896068438535915623091, 3.16080152176289073528751699268, 3.55177581344733015412600312264, 4.56440343433289668921138084062, 5.18216684769054257787794547017, 5.84126322781191730929599779377, 6.57058508708319604080006445584, 7.57825594242192797000800499426, 7.85142846098016081198835168386
