L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s + 4·11-s − 12-s + 13-s − 2·15-s + 16-s + 2·17-s + 18-s − 4·19-s + 2·20-s + 4·22-s + 4·23-s − 24-s − 25-s + 26-s − 27-s − 2·30-s + 4·31-s + 32-s − 4·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.365·30-s + 0.718·31-s + 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.402719386\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.402719386\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.555950135770153175722078590756, −7.51259176044764923376355013063, −6.69976829728960025795025591878, −6.13600376185177616695611061811, −5.67769924839071008086309403991, −4.68660652125533768074019547190, −4.07263049223616849086736384490, −3.04676119237988254824486643336, −1.96531961081237059088925332430, −1.07009503768608329284422606554,
1.07009503768608329284422606554, 1.96531961081237059088925332430, 3.04676119237988254824486643336, 4.07263049223616849086736384490, 4.68660652125533768074019547190, 5.67769924839071008086309403991, 6.13600376185177616695611061811, 6.69976829728960025795025591878, 7.51259176044764923376355013063, 8.555950135770153175722078590756