| L(s) = 1 | + 28.2·7-s + 47.8·11-s + 10.9·13-s + 28.6·17-s + 86.4·19-s + 49.2·23-s + 165.·29-s − 247.·31-s + 375.·37-s + 504.·41-s − 207.·43-s − 70.1·47-s + 455.·49-s − 286.·53-s − 92.3·59-s − 697.·61-s − 840.·67-s − 303.·71-s − 251.·73-s + 1.35e3·77-s + 745.·79-s + 1.32e3·83-s − 415.·89-s + 308.·91-s + 656.·97-s − 1.22e3·101-s − 598.·103-s + ⋯ |
| L(s) = 1 | + 1.52·7-s + 1.31·11-s + 0.233·13-s + 0.408·17-s + 1.04·19-s + 0.446·23-s + 1.05·29-s − 1.43·31-s + 1.66·37-s + 1.92·41-s − 0.737·43-s − 0.217·47-s + 1.32·49-s − 0.742·53-s − 0.203·59-s − 1.46·61-s − 1.53·67-s − 0.507·71-s − 0.403·73-s + 2.00·77-s + 1.06·79-s + 1.75·83-s − 0.495·89-s + 0.355·91-s + 0.687·97-s − 1.20·101-s − 0.572·103-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
| \(L(2)\) |
\(\approx\) |
\(3.490587901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.490587901\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 28.2T + 343T^{2} \) |
| 11 | \( 1 - 47.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 375.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 504.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 70.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 286.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 92.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 697.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 840.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 303.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 745.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 415.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 656.T + 9.12e5T^{2} \) |
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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
−9.055819543235283651551337662546, −7.965301959184464216712135645924, −7.56665520716847666025757328155, −6.50653982858942303644877911399, −5.64709342317520192841882218821, −4.76110100606079401828672490245, −4.06025238040914543547088186750, −2.94565573583683575570587901244, −1.61367365597854208747869411818, −1.00396716728064498550190398422,
1.00396716728064498550190398422, 1.61367365597854208747869411818, 2.94565573583683575570587901244, 4.06025238040914543547088186750, 4.76110100606079401828672490245, 5.64709342317520192841882218821, 6.50653982858942303644877911399, 7.56665520716847666025757328155, 7.965301959184464216712135645924, 9.055819543235283651551337662546
