L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s − 7-s + 8-s + 9-s + 3·10-s + 11-s − 12-s + 13-s − 14-s − 3·15-s + 16-s − 17-s + 18-s + 6·19-s + 3·20-s + 21-s + 22-s − 2·23-s − 24-s + 4·25-s + 26-s − 27-s − 28-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 + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.37·19-s + 0.670·20-s + 0.218·21-s + 0.213·22-s − 0.417·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.434564338\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.434564338\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−10.34267925432200827607502244590, −9.808333759097810918139633537630, −8.922264395526654548902469503440, −7.49197182645084042752453914012, −6.56659909200568568766457956335, −5.84903252742521504807352083794, −5.24787497861869108766000247667, −4.02034094062438424710166423761, −2.73638665036642597233467391828, −1.43984798308906426869123656513,
1.43984798308906426869123656513, 2.73638665036642597233467391828, 4.02034094062438424710166423761, 5.24787497861869108766000247667, 5.84903252742521504807352083794, 6.56659909200568568766457956335, 7.49197182645084042752453914012, 8.922264395526654548902469503440, 9.808333759097810918139633537630, 10.34267925432200827607502244590