| L(s) = 1 | − 2·3-s + 4·5-s − 4·7-s + 9-s − 2·11-s − 2·13-s − 8·15-s + 8·19-s + 8·21-s − 8·23-s + 11·25-s + 4·27-s + 4·29-s − 4·31-s + 4·33-s − 16·35-s − 8·37-s + 4·39-s + 2·41-s − 4·43-s + 4·45-s + 9·49-s + 6·53-s − 8·55-s − 16·57-s + 4·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 2.06·15-s + 1.83·19-s + 1.74·21-s − 1.66·23-s + 11/5·25-s + 0.769·27-s + 0.742·29-s − 0.718·31-s + 0.696·33-s − 2.70·35-s − 1.31·37-s + 0.640·39-s + 0.312·41-s − 0.609·43-s + 0.596·45-s + 9/7·49-s + 0.824·53-s − 1.07·55-s − 2.11·57-s + 0.520·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.034705644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034705644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.04198172776519, −15.48367005085555, −14.41010850983111, −14.02313288683459, −13.53203311902424, −13.02370082153438, −12.41896117462065, −12.04664096914133, −11.37365395568305, −10.43685282685606, −10.17028687229628, −9.818304412238182, −9.299799967278100, −8.578373802464529, −7.581558687421798, −6.766612745619773, −6.539099540114557, −5.736507404304361, −5.463918402047706, −5.066270570228452, −3.867871746635131, −2.967404562498615, −2.460328866426307, −1.482222160118405, −0.4541066944649153,
0.4541066944649153, 1.482222160118405, 2.460328866426307, 2.967404562498615, 3.867871746635131, 5.066270570228452, 5.463918402047706, 5.736507404304361, 6.539099540114557, 6.766612745619773, 7.581558687421798, 8.578373802464529, 9.299799967278100, 9.818304412238182, 10.17028687229628, 10.43685282685606, 11.37365395568305, 12.04664096914133, 12.41896117462065, 13.02370082153438, 13.53203311902424, 14.02313288683459, 14.41010850983111, 15.48367005085555, 16.04198172776519
