| L(s) = 1 | + 0.447·2-s − 9.51·3-s − 7.79·4-s + 11.7·5-s − 4.25·6-s − 2.27·7-s − 7.07·8-s + 63.5·9-s + 5.25·10-s − 22.6·11-s + 74.2·12-s + 67.4·13-s − 1.01·14-s − 111.·15-s + 59.2·16-s + 28.4·18-s + 42.9·19-s − 91.5·20-s + 21.6·21-s − 10.1·22-s − 117.·23-s + 67.2·24-s + 12.8·25-s + 30.1·26-s − 347.·27-s + 17.7·28-s − 226.·29-s + ⋯ |
| L(s) = 1 | + 0.158·2-s − 1.83·3-s − 0.974·4-s + 1.05·5-s − 0.289·6-s − 0.122·7-s − 0.312·8-s + 2.35·9-s + 0.166·10-s − 0.621·11-s + 1.78·12-s + 1.43·13-s − 0.0194·14-s − 1.92·15-s + 0.925·16-s + 0.372·18-s + 0.519·19-s − 1.02·20-s + 0.225·21-s − 0.0984·22-s − 1.06·23-s + 0.572·24-s + 0.103·25-s + 0.227·26-s − 2.47·27-s + 0.119·28-s − 1.44·29-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 | \( 1 - 0.447T + 8T^{2} \) |
| 3 | \( 1 + 9.51T + 27T^{2} \) |
| 5 | \( 1 - 11.7T + 125T^{2} \) |
| 7 | \( 1 + 2.27T + 343T^{2} \) |
| 11 | \( 1 + 22.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.4T + 2.19e3T^{2} \) |
| 19 | \( 1 - 42.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 117.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 0.673T + 2.97e4T^{2} \) |
| 37 | \( 1 - 99.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 30.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 147.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 321.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 612.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 248.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 701.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 773.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.64e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 479.T + 9.12e5T^{2} \) |
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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
−10.89955303916345664248739244693, −10.04948337090509513074947304436, −9.383471077972960155094435873882, −7.925889336726149091420349397719, −6.37743745176777619964029315071, −5.76660402676453008942502677961, −5.10203093462455298518573394664, −3.86104316188538243498343269302, −1.42765235796776634997395105303, 0,
1.42765235796776634997395105303, 3.86104316188538243498343269302, 5.10203093462455298518573394664, 5.76660402676453008942502677961, 6.37743745176777619964029315071, 7.925889336726149091420349397719, 9.383471077972960155094435873882, 10.04948337090509513074947304436, 10.89955303916345664248739244693
