L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s + 6·11-s − 12-s + 13-s − 4·14-s + 16-s + 6·17-s − 18-s − 19-s − 4·21-s − 6·22-s + 6·23-s + 24-s − 26-s − 27-s + 4·28-s − 3·29-s + 2·31-s − 32-s − 6·33-s − 6·34-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 + 1.80·11-s − 0.288·12-s + 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.229·19-s − 0.872·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s − 1.04·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.487479104\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.487479104\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.29487468376526, −14.82514413411565, −14.34855730664147, −14.09762115200207, −13.06111666929004, −12.50629851264189, −11.76388332666922, −11.63657439963542, −11.02881887287994, −10.64385694910998, −9.823911192314952, −9.294805360083772, −8.825188076697629, −8.130887353491862, −7.609891604879021, −7.101898901876357, −6.332285898604873, −5.870807171381042, −5.124838836071511, −4.489694962832662, −3.844078522271964, −3.016467713078180, −1.905156205675256, −1.255037071984660, −0.8852570599653922,
0.8852570599653922, 1.255037071984660, 1.905156205675256, 3.016467713078180, 3.844078522271964, 4.489694962832662, 5.124838836071511, 5.870807171381042, 6.332285898604873, 7.101898901876357, 7.609891604879021, 8.130887353491862, 8.825188076697629, 9.294805360083772, 9.823911192314952, 10.64385694910998, 11.02881887287994, 11.63657439963542, 11.76388332666922, 12.50629851264189, 13.06111666929004, 14.09762115200207, 14.34855730664147, 14.82514413411565, 15.29487468376526