L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s − 2·9-s − 12-s + 5·13-s + 4·14-s + 16-s − 2·18-s − 19-s − 4·21-s − 3·23-s − 24-s − 5·25-s + 5·26-s + 5·27-s + 4·28-s − 9·29-s + 4·31-s + 32-s − 2·36-s − 5·37-s − 38-s − 5·39-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 − 2/3·9-s − 0.288·12-s + 1.38·13-s + 1.06·14-s + 1/4·16-s − 0.471·18-s − 0.229·19-s − 0.872·21-s − 0.625·23-s − 0.204·24-s − 25-s + 0.980·26-s + 0.962·27-s + 0.755·28-s − 1.67·29-s + 0.718·31-s + 0.176·32-s − 1/3·36-s − 0.821·37-s − 0.162·38-s − 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30634 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30634 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.328361467\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.328361467\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 17 | \( 1 \) |
| 53 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.15651451045117, −14.54281076757294, −13.92283269714195, −13.68440905751537, −13.13255234918457, −12.29940678523219, −11.67336947483450, −11.62080391946523, −10.97292671552269, −10.59817190210545, −9.949521667682092, −8.900591563886061, −8.586797230749578, −7.887775624839132, −7.547225982923380, −6.453871529056328, −6.236179072438306, −5.351062824121115, −5.250605058142368, −4.395113669806877, −3.772864918344713, −3.207131220118186, −1.972792988425201, −1.767633254508646, −0.6346903723477665,
0.6346903723477665, 1.767633254508646, 1.972792988425201, 3.207131220118186, 3.772864918344713, 4.395113669806877, 5.250605058142368, 5.351062824121115, 6.236179072438306, 6.453871529056328, 7.547225982923380, 7.887775624839132, 8.586797230749578, 8.900591563886061, 9.949521667682092, 10.59817190210545, 10.97292671552269, 11.62080391946523, 11.67336947483450, 12.29940678523219, 13.13255234918457, 13.68440905751537, 13.92283269714195, 14.54281076757294, 15.15651451045117