| L(s) = 1 | − 2.78·2-s − 3·3-s − 0.253·4-s + 1.24·5-s + 8.34·6-s + 31.9·7-s + 22.9·8-s + 9·9-s − 3.45·10-s + 11·11-s + 0.760·12-s − 83.9·13-s − 88.8·14-s − 3.72·15-s − 61.9·16-s − 118.·17-s − 25.0·18-s − 57.4·19-s − 0.314·20-s − 95.7·21-s − 30.6·22-s + 157.·23-s − 68.9·24-s − 123.·25-s + 233.·26-s − 27·27-s − 8.09·28-s + ⋯ |
| L(s) = 1 | − 0.984·2-s − 0.577·3-s − 0.0317·4-s + 0.111·5-s + 0.568·6-s + 1.72·7-s + 1.01·8-s + 0.333·9-s − 0.109·10-s + 0.301·11-s + 0.0183·12-s − 1.79·13-s − 1.69·14-s − 0.0640·15-s − 0.967·16-s − 1.68·17-s − 0.328·18-s − 0.693·19-s − 0.00351·20-s − 0.994·21-s − 0.296·22-s + 1.42·23-s − 0.586·24-s − 0.987·25-s + 1.76·26-s − 0.192·27-s − 0.0546·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7564642597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7564642597\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 2.78T + 8T^{2} \) |
| 5 | \( 1 - 1.24T + 125T^{2} \) |
| 7 | \( 1 - 31.9T + 343T^{2} \) |
| 13 | \( 1 + 83.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 57.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 85.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 3.02T + 2.97e4T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 266.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 98.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 33.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 343.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 152.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 732.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 751.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 478.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 50.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 576.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 849.T + 9.12e5T^{2} \) |
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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
−8.861542841216600358178874532483, −8.088641959301490260471604755127, −7.36179910276858738861470556063, −6.77463076607293185343333595030, −5.42277344300422705899152255159, −4.64011628526319137252686327406, −4.38381469986298692720611272490, −2.33572695124761605859538690411, −1.64819884276033383268461174577, −0.49094102702717226801476829500,
0.49094102702717226801476829500, 1.64819884276033383268461174577, 2.33572695124761605859538690411, 4.38381469986298692720611272490, 4.64011628526319137252686327406, 5.42277344300422705899152255159, 6.77463076607293185343333595030, 7.36179910276858738861470556063, 8.088641959301490260471604755127, 8.861542841216600358178874532483
