| L(s) = 1 | + 3.67·2-s + 5.47·4-s + 21.0·7-s − 9.28·8-s − 11·11-s + 65.3·13-s + 77.2·14-s − 77.8·16-s − 58.8·17-s − 48.3·19-s − 40.3·22-s + 74.3·23-s + 239.·26-s + 115.·28-s − 17.8·29-s + 324.·31-s − 211.·32-s − 216.·34-s + 38.7·37-s − 177.·38-s + 372.·41-s − 459.·43-s − 60.1·44-s + 273.·46-s + 558.·47-s + 99.8·49-s + 357.·52-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.683·4-s + 1.13·7-s − 0.410·8-s − 0.301·11-s + 1.39·13-s + 1.47·14-s − 1.21·16-s − 0.840·17-s − 0.583·19-s − 0.391·22-s + 0.674·23-s + 1.80·26-s + 0.777·28-s − 0.114·29-s + 1.88·31-s − 1.16·32-s − 1.09·34-s + 0.172·37-s − 0.757·38-s + 1.41·41-s − 1.62·43-s − 0.206·44-s + 0.875·46-s + 1.73·47-s + 0.291·49-s + 0.952·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.215519681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.215519681\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.67T + 8T^{2} \) |
| 7 | \( 1 - 21.0T + 343T^{2} \) |
| 13 | \( 1 - 65.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 17.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 324.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 38.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 372.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 459.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 558.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 363.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 263.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 385.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 647.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 759.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 707.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 879.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 380.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 149.T + 9.12e5T^{2} \) |
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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
−8.591339370379980324648770987557, −7.81709754601812232764253350507, −6.70755002502217171366539163513, −6.11123444042289298524282135506, −5.29070645926578896295269780749, −4.53719710983250473824993275558, −4.03345478002783689797138613965, −2.96355068776382631687959330456, −2.06162858081162675042700083362, −0.841313751329456218349128215169,
0.841313751329456218349128215169, 2.06162858081162675042700083362, 2.96355068776382631687959330456, 4.03345478002783689797138613965, 4.53719710983250473824993275558, 5.29070645926578896295269780749, 6.11123444042289298524282135506, 6.70755002502217171366539163513, 7.81709754601812232764253350507, 8.591339370379980324648770987557
