| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 5·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 10·10-s − 60·11-s + 12·12-s − 64·13-s − 14·14-s + 15·15-s + 16·16-s − 129·17-s + 18·18-s − 52·19-s + 20·20-s − 21·21-s − 120·22-s − 23·23-s + 24·24-s + 25·25-s − 128·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.64·11-s + 0.288·12-s − 1.36·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.84·17-s + 0.235·18-s − 0.627·19-s + 0.223·20-s − 0.218·21-s − 1.16·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.965·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 + p T \) |
| good | 7 | \( 1 + p T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 64 T + p^{3} T^{2} \) |
| 17 | \( 1 + 129 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 99 T + p^{3} T^{2} \) |
| 31 | \( 1 + 115 T + p^{3} T^{2} \) |
| 37 | \( 1 - 137 T + p^{3} T^{2} \) |
| 41 | \( 1 - 327 T + p^{3} T^{2} \) |
| 43 | \( 1 - 500 T + p^{3} T^{2} \) |
| 47 | \( 1 + 258 T + p^{3} T^{2} \) |
| 53 | \( 1 - 555 T + p^{3} T^{2} \) |
| 59 | \( 1 - 471 T + p^{3} T^{2} \) |
| 61 | \( 1 - 614 T + p^{3} T^{2} \) |
| 67 | \( 1 + 307 T + p^{3} T^{2} \) |
| 71 | \( 1 + 627 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1072 T + p^{3} T^{2} \) |
| 79 | \( 1 - 692 T + p^{3} T^{2} \) |
| 83 | \( 1 - 903 T + p^{3} T^{2} \) |
| 89 | \( 1 + 528 T + p^{3} T^{2} \) |
| 97 | \( 1 + 250 T + p^{3} T^{2} \) |
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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
−9.710504466382981937214971495348, −8.814906439621215163454943608959, −7.70806156718939541107829360853, −7.02704656725319971586634033418, −5.93237787479776130709033810175, −4.97659374055700733517230497973, −4.10668327923604428118784350071, −2.57024193531667093836892745726, −2.30452225355940210474870434016, 0,
2.30452225355940210474870434016, 2.57024193531667093836892745726, 4.10668327923604428118784350071, 4.97659374055700733517230497973, 5.93237787479776130709033810175, 7.02704656725319971586634033418, 7.70806156718939541107829360853, 8.814906439621215163454943608959, 9.710504466382981937214971495348
