| L(s) = 1 | + 2.78·2-s − 3.55·3-s − 0.244·4-s + 9.98·5-s − 9.89·6-s − 22.9·8-s − 14.3·9-s + 27.8·10-s − 46.6·11-s + 0.869·12-s + 47.2·13-s − 35.4·15-s − 61.9·16-s − 8.20·17-s − 40.0·18-s + 118.·19-s − 2.44·20-s − 129.·22-s + 183.·23-s + 81.5·24-s − 25.2·25-s + 131.·26-s + 146.·27-s + 223.·29-s − 98.7·30-s − 130.·31-s + 11.0·32-s + ⋯ |
| L(s) = 1 | + 0.984·2-s − 0.683·3-s − 0.0305·4-s + 0.893·5-s − 0.673·6-s − 1.01·8-s − 0.532·9-s + 0.879·10-s − 1.27·11-s + 0.0209·12-s + 1.00·13-s − 0.610·15-s − 0.968·16-s − 0.117·17-s − 0.524·18-s + 1.43·19-s − 0.0273·20-s − 1.25·22-s + 1.66·23-s + 0.693·24-s − 0.202·25-s + 0.992·26-s + 1.04·27-s + 1.43·29-s − 0.601·30-s − 0.758·31-s + 0.0611·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 - 2.78T + 8T^{2} \) |
| 3 | \( 1 + 3.55T + 27T^{2} \) |
| 5 | \( 1 - 9.98T + 125T^{2} \) |
| 11 | \( 1 + 46.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 47.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 8.20T + 4.91e3T^{2} \) |
| 19 | \( 1 - 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 130.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20.7T + 5.06e4T^{2} \) |
| 43 | \( 1 - 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 544.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 369.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 563.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 173.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 964.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 655.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 924.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 291.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 22.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 185.T + 9.12e5T^{2} \) |
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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
−8.545925860162437214335819260295, −7.45356429121214024118162799410, −6.32586309017264768719728928380, −5.86302195813472161769704898479, −5.16980315871176009017583275202, −4.72664690459749093445747649761, −3.24136871209465192779296596699, −2.80453396167735813192387474265, −1.24872764133827816112769870674, 0,
1.24872764133827816112769870674, 2.80453396167735813192387474265, 3.24136871209465192779296596699, 4.72664690459749093445747649761, 5.16980315871176009017583275202, 5.86302195813472161769704898479, 6.32586309017264768719728928380, 7.45356429121214024118162799410, 8.545925860162437214335819260295
