| L(s) = 1 | − 1.58·2-s − 4.98·3-s − 5.49·4-s − 14.4·5-s + 7.88·6-s + 29.4·7-s + 21.3·8-s − 2.16·9-s + 22.8·10-s − 13.5·11-s + 27.4·12-s + 45.7·13-s − 46.5·14-s + 72.0·15-s + 10.2·16-s + 3.41·18-s + 113.·19-s + 79.4·20-s − 146.·21-s + 21.4·22-s − 144.·23-s − 106.·24-s + 83.7·25-s − 72.3·26-s + 145.·27-s − 161.·28-s + 1.26·29-s + ⋯ |
| L(s) = 1 | − 0.559·2-s − 0.959·3-s − 0.687·4-s − 1.29·5-s + 0.536·6-s + 1.58·7-s + 0.943·8-s − 0.0800·9-s + 0.722·10-s − 0.370·11-s + 0.659·12-s + 0.976·13-s − 0.888·14-s + 1.23·15-s + 0.159·16-s + 0.0447·18-s + 1.36·19-s + 0.888·20-s − 1.52·21-s + 0.207·22-s − 1.31·23-s − 0.905·24-s + 0.669·25-s − 0.546·26-s + 1.03·27-s − 1.09·28-s + 0.00811·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 1.58T + 8T^{2} \) |
| 3 | \( 1 + 4.98T + 27T^{2} \) |
| 5 | \( 1 + 14.4T + 125T^{2} \) |
| 7 | \( 1 - 29.4T + 343T^{2} \) |
| 11 | \( 1 + 13.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 1.26T + 2.43e4T^{2} \) |
| 31 | \( 1 - 30.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 398.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 135.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 635.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 625.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 166.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 159.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 19.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 336.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 47.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 626.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 692.T + 9.12e5T^{2} \) |
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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
−11.08019424357732429019464087430, −10.17879190970602887476858044314, −8.631497477324700614442224030894, −8.151123025019016515318299641886, −7.32970152747774403772024660141, −5.59554661164460924808695712358, −4.80759154398092360658069768454, −3.78045879569925759419261907650, −1.26153218601736440864435994908, 0,
1.26153218601736440864435994908, 3.78045879569925759419261907650, 4.80759154398092360658069768454, 5.59554661164460924808695712358, 7.32970152747774403772024660141, 8.151123025019016515318299641886, 8.631497477324700614442224030894, 10.17879190970602887476858044314, 11.08019424357732429019464087430
