| L(s) = 1 | − 23.9·7-s + 22.9·11-s + 60.8·13-s − 26.9·17-s − 37.7·19-s + 16.7·23-s − 306.·29-s + 109.·31-s + 179.·37-s − 66.3·41-s + 383.·43-s − 270.·47-s + 228.·49-s + 54.3·53-s + 213.·59-s − 732.·61-s + 1.05e3·67-s + 1.02e3·71-s + 591.·73-s − 547.·77-s + 66.1·79-s − 619.·83-s − 1.00e3·89-s − 1.45e3·91-s − 606.·97-s − 412.·101-s − 1.18e3·103-s + ⋯ |
| L(s) = 1 | − 1.29·7-s + 0.627·11-s + 1.29·13-s − 0.383·17-s − 0.455·19-s + 0.151·23-s − 1.96·29-s + 0.633·31-s + 0.797·37-s − 0.252·41-s + 1.35·43-s − 0.839·47-s + 0.666·49-s + 0.140·53-s + 0.470·59-s − 1.53·61-s + 1.92·67-s + 1.71·71-s + 0.947·73-s − 0.810·77-s + 0.0942·79-s − 0.818·83-s − 1.19·89-s − 1.67·91-s − 0.634·97-s − 0.406·101-s − 1.12·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 23.9T + 343T^{2} \) |
| 11 | \( 1 - 22.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 26.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 16.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 306.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 109.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 66.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 270.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 54.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 213.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 732.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 591.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 66.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 619.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 606.T + 9.12e5T^{2} \) |
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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
−8.659200095688868914838211273992, −7.74596938661892142430171237271, −6.65650270458210145966859831172, −6.32072133532544498638575214906, −5.43198688167205370744747466064, −4.06942302483626015670741981293, −3.59817969884900890684348407779, −2.48761835565738179474531515071, −1.21511692913021293780627915718, 0,
1.21511692913021293780627915718, 2.48761835565738179474531515071, 3.59817969884900890684348407779, 4.06942302483626015670741981293, 5.43198688167205370744747466064, 6.32072133532544498638575214906, 6.65650270458210145966859831172, 7.74596938661892142430171237271, 8.659200095688868914838211273992
