L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s + 7-s − 8-s + 9-s − 3·10-s + 4·11-s − 12-s + 3·13-s − 14-s − 3·15-s + 16-s − 18-s + 3·20-s − 21-s − 4·22-s − 23-s + 24-s + 4·25-s − 3·26-s − 27-s + 28-s + 29-s + 3·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.20·11-s − 0.288·12-s + 0.832·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.670·20-s − 0.218·21-s − 0.852·22-s − 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.588·26-s − 0.192·27-s + 0.188·28-s + 0.185·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.409709048\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409709048\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−10.03966127807896462669928239171, −9.189845448951089390655711400514, −8.655745092843968444463701065709, −7.43185206340477597298102477560, −6.42286204202105611639154452768, −6.02017693542824857854578311692, −4.98434482137604303899714629871, −3.65110433129821651966310469603, −2.04863164245660824135306206527, −1.19440309177586005149491227609,
1.19440309177586005149491227609, 2.04863164245660824135306206527, 3.65110433129821651966310469603, 4.98434482137604303899714629871, 6.02017693542824857854578311692, 6.42286204202105611639154452768, 7.43185206340477597298102477560, 8.655745092843968444463701065709, 9.189845448951089390655711400514, 10.03966127807896462669928239171