| L(s) = 1 | − 4.59·2-s + 13.1·4-s − 5·5-s + 20.6·7-s − 23.4·8-s + 22.9·10-s − 11·11-s − 15.6·13-s − 94.8·14-s + 3.04·16-s − 72.9·17-s + 61.0·19-s − 65.5·20-s + 50.5·22-s + 13.6·23-s + 25·25-s + 71.9·26-s + 270.·28-s + 31.4·29-s − 243.·31-s + 173.·32-s + 335.·34-s − 103.·35-s − 65.4·37-s − 280.·38-s + 117.·40-s + 109.·41-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.63·4-s − 0.447·5-s + 1.11·7-s − 1.03·8-s + 0.726·10-s − 0.301·11-s − 0.334·13-s − 1.81·14-s + 0.0475·16-s − 1.04·17-s + 0.737·19-s − 0.733·20-s + 0.489·22-s + 0.123·23-s + 0.200·25-s + 0.542·26-s + 1.82·28-s + 0.201·29-s − 1.40·31-s + 0.961·32-s + 1.69·34-s − 0.498·35-s − 0.290·37-s − 1.19·38-s + 0.464·40-s + 0.415·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.59T + 8T^{2} \) |
| 7 | \( 1 - 20.6T + 343T^{2} \) |
| 13 | \( 1 + 15.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 31.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 109.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 121.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 519.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 542.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 109.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 89.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 488.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 837.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 351.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 831.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 426.T + 9.12e5T^{2} \) |
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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
−10.02239973280392146058121914907, −9.012208843607188789257948988859, −8.428513078819952736361945998923, −7.53576274164968424666034013639, −7.00551041545507967372726895205, −5.44839872283170117986043361097, −4.26889277569944493069660726769, −2.50954114533902255868507922190, −1.36094543930977937555472936327, 0,
1.36094543930977937555472936327, 2.50954114533902255868507922190, 4.26889277569944493069660726769, 5.44839872283170117986043361097, 7.00551041545507967372726895205, 7.53576274164968424666034013639, 8.428513078819952736361945998923, 9.012208843607188789257948988859, 10.02239973280392146058121914907
