| L(s) = 1 | + 3.67·2-s + 3·3-s + 5.48·4-s + 4.96·5-s + 11.0·6-s + 33.5·7-s − 9.22·8-s + 9·9-s + 18.2·10-s + 11·11-s + 16.4·12-s + 16.0·13-s + 123.·14-s + 14.8·15-s − 77.7·16-s − 88.2·17-s + 33.0·18-s + 87.1·19-s + 27.2·20-s + 100.·21-s + 40.3·22-s + 150.·23-s − 27.6·24-s − 100.·25-s + 59.0·26-s + 27·27-s + 183.·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.577·3-s + 0.685·4-s + 0.443·5-s + 0.749·6-s + 1.80·7-s − 0.407·8-s + 0.333·9-s + 0.576·10-s + 0.301·11-s + 0.395·12-s + 0.342·13-s + 2.35·14-s + 0.256·15-s − 1.21·16-s − 1.25·17-s + 0.432·18-s + 1.05·19-s + 0.304·20-s + 1.04·21-s + 0.391·22-s + 1.36·23-s − 0.235·24-s − 0.802·25-s + 0.445·26-s + 0.192·27-s + 1.24·28-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
| \(L(2)\) |
\(\approx\) |
\(7.967388121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.967388121\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.67T + 8T^{2} \) |
| 5 | \( 1 - 4.96T + 125T^{2} \) |
| 7 | \( 1 - 33.5T + 343T^{2} \) |
| 13 | \( 1 - 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 88.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 247.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 93.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 444.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 591.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 498.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 819.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 208.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 202.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 81.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 156.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 573.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 67.7T + 9.12e5T^{2} \) |
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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
−8.837087847960333845614365058194, −7.991503188408373753915397972109, −7.12910744236647935474734596099, −6.22864321187175143772783181331, −5.30138570613986856830488915963, −4.70426521356469440697720189629, −4.08840985546756941624341477929, −2.97455653881856338637619086349, −2.12321580976128710633806461491, −1.13663663706570936679153571911,
1.13663663706570936679153571911, 2.12321580976128710633806461491, 2.97455653881856338637619086349, 4.08840985546756941624341477929, 4.70426521356469440697720189629, 5.30138570613986856830488915963, 6.22864321187175143772783181331, 7.12910744236647935474734596099, 7.991503188408373753915397972109, 8.837087847960333845614365058194
