| L(s) = 1 | − 5.35·2-s + 0.409·3-s + 20.6·4-s − 12.3·5-s − 2.19·6-s + 3.32·7-s − 67.7·8-s − 26.8·9-s + 65.9·10-s + 8.46·12-s − 13·13-s − 17.7·14-s − 5.04·15-s + 197.·16-s + 23.1·17-s + 143.·18-s − 6.99·19-s − 254.·20-s + 1.36·21-s − 173.·23-s − 27.7·24-s + 26.6·25-s + 69.5·26-s − 22.0·27-s + 68.6·28-s + 133.·29-s + 26.9·30-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.0788·3-s + 2.58·4-s − 1.10·5-s − 0.149·6-s + 0.179·7-s − 2.99·8-s − 0.993·9-s + 2.08·10-s + 0.203·12-s − 0.277·13-s − 0.339·14-s − 0.0868·15-s + 3.08·16-s + 0.330·17-s + 1.88·18-s − 0.0844·19-s − 2.84·20-s + 0.0141·21-s − 1.57·23-s − 0.235·24-s + 0.212·25-s + 0.524·26-s − 0.157·27-s + 0.463·28-s + 0.855·29-s + 0.164·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 5.35T + 8T^{2} \) |
| 3 | \( 1 - 0.409T + 27T^{2} \) |
| 5 | \( 1 + 12.3T + 125T^{2} \) |
| 7 | \( 1 - 3.32T + 343T^{2} \) |
| 17 | \( 1 - 23.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.99T + 6.85e3T^{2} \) |
| 23 | \( 1 + 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 392.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 369.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 317.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 22.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 552.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 274.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 53.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 440.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 704.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 68.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−8.446888090891534348305044312330, −8.019876176731526538173308883203, −7.61444444481030155561282561198, −6.51746877910376507713973932503, −5.80468241560184204091186369787, −4.28685087741244934879286402652, −3.07144567905366072881408481318, −2.24065445313680939355492714001, −0.866905811384094475537013575950, 0,
0.866905811384094475537013575950, 2.24065445313680939355492714001, 3.07144567905366072881408481318, 4.28685087741244934879286402652, 5.80468241560184204091186369787, 6.51746877910376507713973932503, 7.61444444481030155561282561198, 8.019876176731526538173308883203, 8.446888090891534348305044312330
