| L(s) = 1 | − 0.449·2-s − 3·3-s − 7.79·4-s + 5.67·5-s + 1.34·6-s + 0.979·7-s + 7.09·8-s + 9·9-s − 2.54·10-s − 11·11-s + 23.3·12-s − 33.6·13-s − 0.439·14-s − 17.0·15-s + 59.1·16-s − 99.2·17-s − 4.04·18-s + 14.5·19-s − 44.2·20-s − 2.93·21-s + 4.94·22-s + 12.6·23-s − 21.2·24-s − 92.8·25-s + 15.1·26-s − 27·27-s − 7.63·28-s + ⋯ |
| L(s) = 1 | − 0.158·2-s − 0.577·3-s − 0.974·4-s + 0.507·5-s + 0.0916·6-s + 0.0528·7-s + 0.313·8-s + 0.333·9-s − 0.0805·10-s − 0.301·11-s + 0.562·12-s − 0.718·13-s − 0.00839·14-s − 0.292·15-s + 0.924·16-s − 1.41·17-s − 0.0529·18-s + 0.175·19-s − 0.494·20-s − 0.0305·21-s + 0.0478·22-s + 0.114·23-s − 0.181·24-s − 0.742·25-s + 0.114·26-s − 0.192·27-s − 0.0515·28-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
| \(L(2)\) |
\(\approx\) |
\(0.7055814837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7055814837\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.449T + 8T^{2} \) |
| 5 | \( 1 - 5.67T + 125T^{2} \) |
| 7 | \( 1 - 0.979T + 343T^{2} \) |
| 13 | \( 1 + 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 12.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 48.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 61.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 284.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 37.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 265.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 438.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 312.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 926.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 526.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 528.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−9.098958411089538295599705470408, −7.954069136521729873733349433096, −7.36793659232646019143107677825, −6.21783321339020698317383552809, −5.63645992604731542724652319858, −4.66164340384806859369274590050, −4.23204749656263987721203044833, −2.80626792380163950503032661846, −1.67567172771617501350786239565, −0.41139967293193779292071569869,
0.41139967293193779292071569869, 1.67567172771617501350786239565, 2.80626792380163950503032661846, 4.23204749656263987721203044833, 4.66164340384806859369274590050, 5.63645992604731542724652319858, 6.21783321339020698317383552809, 7.36793659232646019143107677825, 7.954069136521729873733349433096, 9.098958411089538295599705470408
