| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 2·7-s + 9-s + 4·11-s − 4·12-s − 2·13-s + 4·14-s − 4·16-s + 3·17-s + 2·18-s + 6·19-s − 4·21-s + 8·22-s + 3·23-s − 4·26-s + 4·27-s + 4·28-s + 4·29-s + 5·31-s − 8·32-s − 8·33-s + 6·34-s + 2·36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.15·12-s − 0.554·13-s + 1.06·14-s − 16-s + 0.727·17-s + 0.471·18-s + 1.37·19-s − 0.872·21-s + 1.70·22-s + 0.625·23-s − 0.784·26-s + 0.769·27-s + 0.755·28-s + 0.742·29-s + 0.898·31-s − 1.41·32-s − 1.39·33-s + 1.02·34-s + 1/3·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.553595229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.553595229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−10.08381803792975585278125127333, −9.193464947963295543951916740612, −8.017786155448086776451258999819, −6.86805803912072232656655811491, −6.29573417931270915386955255914, −5.23918447401854883211931070345, −5.00372362800030946374427634034, −3.94158192158522125727405547336, −2.85613797509263610359050700873, −1.14774486989900542020378251558,
1.14774486989900542020378251558, 2.85613797509263610359050700873, 3.94158192158522125727405547336, 5.00372362800030946374427634034, 5.23918447401854883211931070345, 6.29573417931270915386955255914, 6.86805803912072232656655811491, 8.017786155448086776451258999819, 9.193464947963295543951916740612, 10.08381803792975585278125127333
