L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 3·9-s − 10-s + 4·11-s − 2·13-s − 4·14-s + 16-s + 17-s − 3·18-s + 19-s − 20-s + 4·22-s + 4·23-s + 25-s − 2·26-s − 4·28-s + 6·29-s − 8·31-s + 32-s + 34-s + 4·35-s − 3·36-s + 6·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 9-s − 0.316·10-s + 1.20·11-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.707·18-s + 0.229·19-s − 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.755·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.171·34-s + 0.676·35-s − 1/2·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.024504729\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.024504729\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−8.891995838972262951043762688480, −7.68356135969199327296404037470, −6.99881592703734884634303699605, −6.32575001694032332508365911527, −5.73741672263566189791307124471, −4.76770086800489199456281077769, −3.77270864544407820344855210884, −3.24427074900902561301248758562, −2.43598927888710964565631726135, −0.74816012003769100428360577929,
0.74816012003769100428360577929, 2.43598927888710964565631726135, 3.24427074900902561301248758562, 3.77270864544407820344855210884, 4.76770086800489199456281077769, 5.73741672263566189791307124471, 6.32575001694032332508365911527, 6.99881592703734884634303699605, 7.68356135969199327296404037470, 8.891995838972262951043762688480