| L(s) = 1 | + 2·3-s − 12.1·5-s + 7·7-s − 23·9-s − 11·11-s + 32.3·13-s − 24.3·15-s − 42.4·17-s + 127.·19-s + 14·21-s + 202.·23-s + 22.9·25-s − 100·27-s + 126.·29-s + 4.16·31-s − 22·33-s − 85.1·35-s − 111.·37-s + 64.6·39-s + 233.·41-s − 476.·43-s + 279.·45-s − 550.·47-s + 49·49-s − 84.9·51-s − 138.·53-s + 133.·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 1.08·5-s + 0.377·7-s − 0.851·9-s − 0.301·11-s + 0.689·13-s − 0.418·15-s − 0.606·17-s + 1.53·19-s + 0.145·21-s + 1.83·23-s + 0.183·25-s − 0.712·27-s + 0.813·29-s + 0.0241·31-s − 0.116·33-s − 0.411·35-s − 0.494·37-s + 0.265·39-s + 0.889·41-s − 1.69·43-s + 0.926·45-s − 1.70·47-s + 0.142·49-s − 0.233·51-s − 0.360·53-s + 0.328·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 2T + 27T^{2} \) |
| 5 | \( 1 + 12.1T + 125T^{2} \) |
| 13 | \( 1 - 32.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 202.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 126.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 4.16T + 2.97e4T^{2} \) |
| 37 | \( 1 + 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 233.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 476.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 550.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 138.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 276.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 619.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 127.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 464.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 103.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 931.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−8.738845106108238624611445375660, −8.181999593543837523931289843218, −7.47796766157947515697691584644, −6.57987725302771674719848964950, −5.38828003209066394005868420055, −4.63260282289843914274474593876, −3.43698223170941542948787903473, −2.89524194664079758459752015036, −1.30494788038398828130851024396, 0,
1.30494788038398828130851024396, 2.89524194664079758459752015036, 3.43698223170941542948787903473, 4.63260282289843914274474593876, 5.38828003209066394005868420055, 6.57987725302771674719848964950, 7.47796766157947515697691584644, 8.181999593543837523931289843218, 8.738845106108238624611445375660
