| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 11·5-s − 6·6-s − 15·7-s − 8·8-s + 9·9-s + 22·10-s − 29·11-s + 12·12-s + 82·13-s + 30·14-s − 33·15-s + 16·16-s + 27·17-s − 18·18-s − 44·20-s − 45·21-s + 58·22-s + 100·23-s − 24·24-s − 4·25-s − 164·26-s + 27·27-s − 60·28-s + 118·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.983·5-s − 0.408·6-s − 0.809·7-s − 0.353·8-s + 1/3·9-s + 0.695·10-s − 0.794·11-s + 0.288·12-s + 1.74·13-s + 0.572·14-s − 0.568·15-s + 1/4·16-s + 0.385·17-s − 0.235·18-s − 0.491·20-s − 0.467·21-s + 0.562·22-s + 0.906·23-s − 0.204·24-s − 0.0319·25-s − 1.23·26-s + 0.192·27-s − 0.404·28-s + 0.755·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.158648313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158648313\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 11 T + p^{3} T^{2} \) |
| 7 | \( 1 + 15 T + p^{3} T^{2} \) |
| 11 | \( 1 + 29 T + p^{3} T^{2} \) |
| 13 | \( 1 - 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 23 | \( 1 - 100 T + p^{3} T^{2} \) |
| 29 | \( 1 - 118 T + p^{3} T^{2} \) |
| 31 | \( 1 + 70 T + p^{3} T^{2} \) |
| 37 | \( 1 + 232 T + p^{3} T^{2} \) |
| 41 | \( 1 + 8 T + p^{3} T^{2} \) |
| 43 | \( 1 + 287 T + p^{3} T^{2} \) |
| 47 | \( 1 - 385 T + p^{3} T^{2} \) |
| 53 | \( 1 + 538 T + p^{3} T^{2} \) |
| 59 | \( 1 - 300 T + p^{3} T^{2} \) |
| 61 | \( 1 + 901 T + p^{3} T^{2} \) |
| 67 | \( 1 + 132 T + p^{3} T^{2} \) |
| 71 | \( 1 + 472 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1131 T + p^{3} T^{2} \) |
| 79 | \( 1 - 52 T + p^{3} T^{2} \) |
| 83 | \( 1 - 276 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1302 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1310 T + p^{3} T^{2} \) |
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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
−8.694850676157717973606416799522, −8.043607600934886828497177522998, −7.40454355439939077812660275746, −6.59343447257804162153435056217, −5.76515426470703171137689321377, −4.51543761825724898645142924560, −3.36740316415758942087018037969, −3.14304564500979714460408797582, −1.65844324157344309486534012342, −0.53220210790183562961394211739,
0.53220210790183562961394211739, 1.65844324157344309486534012342, 3.14304564500979714460408797582, 3.36740316415758942087018037969, 4.51543761825724898645142924560, 5.76515426470703171137689321377, 6.59343447257804162153435056217, 7.40454355439939077812660275746, 8.043607600934886828497177522998, 8.694850676157717973606416799522
