| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·5-s − 6·6-s + 19·7-s + 8·8-s + 9·9-s − 12·10-s + 32·11-s − 12·12-s − 81·13-s + 38·14-s + 18·15-s + 16·16-s − 124·17-s + 18·18-s − 24·20-s − 57·21-s + 64·22-s + 98·23-s − 24·24-s − 89·25-s − 162·26-s − 27·27-s + 76·28-s + 300·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.536·5-s − 0.408·6-s + 1.02·7-s + 0.353·8-s + 1/3·9-s − 0.379·10-s + 0.877·11-s − 0.288·12-s − 1.72·13-s + 0.725·14-s + 0.309·15-s + 1/4·16-s − 1.76·17-s + 0.235·18-s − 0.268·20-s − 0.592·21-s + 0.620·22-s + 0.888·23-s − 0.204·24-s − 0.711·25-s − 1.22·26-s − 0.192·27-s + 0.512·28-s + 1.92·29-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
| \(L(2)\) |
\(\approx\) |
\(2.661588329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.661588329\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 19 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 81 T + p^{3} T^{2} \) |
| 17 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 98 T + p^{3} T^{2} \) |
| 29 | \( 1 - 300 T + p^{3} T^{2} \) |
| 31 | \( 1 - 225 T + p^{3} T^{2} \) |
| 37 | \( 1 - 293 T + p^{3} T^{2} \) |
| 41 | \( 1 + 176 T + p^{3} T^{2} \) |
| 43 | \( 1 + 111 T + p^{3} T^{2} \) |
| 47 | \( 1 + 550 T + p^{3} T^{2} \) |
| 53 | \( 1 - 482 T + p^{3} T^{2} \) |
| 59 | \( 1 - 496 T + p^{3} T^{2} \) |
| 61 | \( 1 - 155 T + p^{3} T^{2} \) |
| 67 | \( 1 + 465 T + p^{3} T^{2} \) |
| 71 | \( 1 - 110 T + p^{3} T^{2} \) |
| 73 | \( 1 - 817 T + p^{3} T^{2} \) |
| 79 | \( 1 + 259 T + p^{3} T^{2} \) |
| 83 | \( 1 + 56 T + p^{3} T^{2} \) |
| 89 | \( 1 + 308 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1150 T + p^{3} T^{2} \) |
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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
−8.520420037308045662710283972966, −7.86667542683077467125042598912, −6.81637737617022093207130826419, −6.59016141760449835006813636941, −5.26277133992078352684711888921, −4.56822255149424366405255484502, −4.31192194170730738808487581312, −2.85048640549290516949262625419, −1.92552648569567870818526430212, −0.68025797810103616577242063494,
0.68025797810103616577242063494, 1.92552648569567870818526430212, 2.85048640549290516949262625419, 4.31192194170730738808487581312, 4.56822255149424366405255484502, 5.26277133992078352684711888921, 6.59016141760449835006813636941, 6.81637737617022093207130826419, 7.86667542683077467125042598912, 8.520420037308045662710283972966
