| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s + 9-s + 11-s + 12-s + 4·13-s + 4·14-s + 16-s + 6·17-s − 18-s − 2·19-s − 4·21-s − 22-s + 6·23-s − 24-s − 5·25-s − 4·26-s + 27-s − 4·28-s + 6·29-s + 4·31-s − 32-s + 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.872·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s − 0.784·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.02903238502315, −13.54991676840129, −13.22889058791077, −12.53305573519356, −12.24803271848296, −11.66627957712726, −11.00080117138557, −10.42889578777893, −10.08346661739730, −9.537164184582753, −9.229814107876597, −8.516189448753757, −8.350123447975704, −7.446847233529855, −7.215584335607894, −6.408435910767971, −6.134096987802285, −5.646027067473370, −4.666050703566204, −3.971117330267493, −3.385799229855488, −2.999431513465089, −2.464967451655467, −1.364931546570562, −1.019921207883743, 0,
1.019921207883743, 1.364931546570562, 2.464967451655467, 2.999431513465089, 3.385799229855488, 3.971117330267493, 4.666050703566204, 5.646027067473370, 6.134096987802285, 6.408435910767971, 7.215584335607894, 7.446847233529855, 8.350123447975704, 8.516189448753757, 9.229814107876597, 9.537164184582753, 10.08346661739730, 10.42889578777893, 11.00080117138557, 11.66627957712726, 12.24803271848296, 12.53305573519356, 13.22889058791077, 13.54991676840129, 14.02903238502315
