| L(s) = 1 | − 1.23·2-s − 3·3-s − 6.46·4-s − 19.0·5-s + 3.71·6-s − 16.0·7-s + 17.9·8-s + 9·9-s + 23.5·10-s + 11·11-s + 19.3·12-s + 20.0·13-s + 19.8·14-s + 57.1·15-s + 29.5·16-s − 41.7·17-s − 11.1·18-s + 14.5·19-s + 123.·20-s + 48.1·21-s − 13.6·22-s + 216.·23-s − 53.7·24-s + 237.·25-s − 24.8·26-s − 27·27-s + 103.·28-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.577·3-s − 0.808·4-s − 1.70·5-s + 0.252·6-s − 0.865·7-s + 0.791·8-s + 0.333·9-s + 0.745·10-s + 0.301·11-s + 0.466·12-s + 0.427·13-s + 0.379·14-s + 0.983·15-s + 0.461·16-s − 0.596·17-s − 0.145·18-s + 0.175·19-s + 1.37·20-s + 0.499·21-s − 0.132·22-s + 1.95·23-s − 0.457·24-s + 1.90·25-s − 0.187·26-s − 0.192·27-s + 0.699·28-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
| \(L(2)\) |
\(\approx\) |
\(0.1081160550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1081160550\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.23T + 8T^{2} \) |
| 5 | \( 1 + 19.0T + 125T^{2} \) |
| 7 | \( 1 + 16.0T + 343T^{2} \) |
| 13 | \( 1 - 20.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 216.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 306.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 73.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 376.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 111.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 96.6T + 2.05e5T^{2} \) |
| 67 | \( 1 + 818.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 607.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 808.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 368.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 316.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 354.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 681.T + 9.12e5T^{2} \) |
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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
−8.894084312044031411531404854564, −8.077554970135830792676110497357, −7.18633385822384240250077501288, −6.78118091033935074827664586467, −5.43645367104481409618412131246, −4.70106054529099340775532866403, −3.76770055561208302996720444000, −3.34158627480506013712035289603, −1.33652163025535357171579700115, −0.18537564944979951758924184546,
0.18537564944979951758924184546, 1.33652163025535357171579700115, 3.34158627480506013712035289603, 3.76770055561208302996720444000, 4.70106054529099340775532866403, 5.43645367104481409618412131246, 6.78118091033935074827664586467, 7.18633385822384240250077501288, 8.077554970135830792676110497357, 8.894084312044031411531404854564
