| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s + 11-s + 2·13-s − 16-s − 6·17-s − 4·19-s + 20-s − 22-s − 4·23-s + 25-s − 2·26-s − 6·29-s − 8·31-s − 5·32-s + 6·34-s − 2·37-s + 4·38-s − 3·40-s − 2·41-s + 4·43-s − 44-s + 4·46-s + 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s − 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s − 1.11·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.474·40-s − 0.312·41-s + 0.609·43-s − 0.150·44-s + 0.589·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.63828874077951155959665793654, −9.301659499748329006626376523886, −8.877176431244694382657420951838, −7.963920428420679646091651044326, −7.06776652987294055072227501878, −5.89992194836808220350577312387, −4.52659700416500112028212971426, −3.79000489498233733500749844862, −1.86919846930486895513918273600, 0,
1.86919846930486895513918273600, 3.79000489498233733500749844862, 4.52659700416500112028212971426, 5.89992194836808220350577312387, 7.06776652987294055072227501878, 7.963920428420679646091651044326, 8.877176431244694382657420951838, 9.301659499748329006626376523886, 10.63828874077951155959665793654
