| L(s) = 1 | − 11.0·2-s + 15.0·3-s + 89.8·4-s − 25·5-s − 166.·6-s − 193.·7-s − 639.·8-s − 15.0·9-s + 276.·10-s + 1.35e3·12-s − 270.·13-s + 2.13e3·14-s − 377.·15-s + 4.17e3·16-s − 1.83e3·17-s + 166.·18-s + 2.27e3·19-s − 2.24e3·20-s − 2.91e3·21-s − 1.86e3·23-s − 9.64e3·24-s + 625·25-s + 2.98e3·26-s − 3.89e3·27-s − 1.73e4·28-s − 843.·29-s + 4.16e3·30-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.968·3-s + 2.80·4-s − 0.447·5-s − 1.89·6-s − 1.49·7-s − 3.53·8-s − 0.0618·9-s + 0.872·10-s + 2.72·12-s − 0.444·13-s + 2.91·14-s − 0.433·15-s + 4.08·16-s − 1.54·17-s + 0.120·18-s + 1.44·19-s − 1.25·20-s − 1.44·21-s − 0.737·23-s − 3.41·24-s + 0.200·25-s + 0.867·26-s − 1.02·27-s − 4.18·28-s − 0.186·29-s + 0.845·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2337789668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2337789668\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 11.0T + 32T^{2} \) |
| 3 | \( 1 - 15.0T + 243T^{2} \) |
| 7 | \( 1 + 193.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 270.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.83e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.86e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 843.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.36e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.80e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.95e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.28e5T + 8.58e9T^{2} \) |
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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
−9.571349562521508281484966497331, −9.052490563343315379840376184055, −8.401506304672308608700918619675, −7.38551531197176930294032211694, −6.91369943749722386388077041510, −5.81815160550963192621018985133, −3.55161862777162997454762585350, −2.84994364319475583454271462243, −1.88446094599018550252677147227, −0.27795792902360342623376866512,
0.27795792902360342623376866512, 1.88446094599018550252677147227, 2.84994364319475583454271462243, 3.55161862777162997454762585350, 5.81815160550963192621018985133, 6.91369943749722386388077041510, 7.38551531197176930294032211694, 8.401506304672308608700918619675, 9.052490563343315379840376184055, 9.571349562521508281484966497331
