L(s) = 1 | − 3-s + 2.30·5-s + 7-s + 9-s − 3.60·11-s − 0.697·13-s − 2.30·15-s − 3·17-s − 7.60·19-s − 21-s − 23-s + 0.302·25-s − 27-s + 29-s + 31-s + 3.60·33-s + 2.30·35-s − 8.21·37-s + 0.697·39-s − 5.60·41-s − 7.90·43-s + 2.30·45-s − 10.6·47-s + 49-s + 3·51-s + 12.9·53-s − 8.30·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.02·5-s + 0.377·7-s + 0.333·9-s − 1.08·11-s − 0.193·13-s − 0.594·15-s − 0.727·17-s − 1.74·19-s − 0.218·21-s − 0.208·23-s + 0.0605·25-s − 0.192·27-s + 0.185·29-s + 0.179·31-s + 0.627·33-s + 0.389·35-s − 1.34·37-s + 0.111·39-s − 0.875·41-s − 1.20·43-s + 0.343·45-s − 1.54·47-s + 0.142·49-s + 0.420·51-s + 1.77·53-s − 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2.30T + 5T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 + 0.697T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 8.21T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 7.90T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8.51T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 + 0.394T + 83T^{2} \) |
| 89 | \( 1 - 3.30T + 89T^{2} \) |
| 97 | \( 1 + 11T + 97T^{2} \) |
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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
−8.651362878125213709904818829322, −8.232838705689800509783592630073, −6.95438563857432411132635324615, −6.46407624520877745263977758138, −5.45265865247141905259912597397, −5.01006435552063346040420928873, −3.95298508279997642186992116669, −2.46817203600319115178278712920, −1.78986418343998711783463871063, 0,
1.78986418343998711783463871063, 2.46817203600319115178278712920, 3.95298508279997642186992116669, 5.01006435552063346040420928873, 5.45265865247141905259912597397, 6.46407624520877745263977758138, 6.95438563857432411132635324615, 8.232838705689800509783592630073, 8.651362878125213709904818829322