| L(s) = 1 | − 2.56·2-s − 1.43·4-s − 6.24·7-s + 24.1·8-s + 11·11-s + 49.1·13-s + 16·14-s − 50.4·16-s + 82.7·17-s − 130.·19-s − 28.1·22-s − 185.·23-s − 125.·26-s + 8.98·28-s + 8.90·29-s + 5.26·31-s − 64.2·32-s − 211.·34-s + 416.·37-s + 333.·38-s + 298.·41-s + 513.·43-s − 15.8·44-s + 475.·46-s + 557.·47-s − 303.·49-s − 70.6·52-s + ⋯ |
| L(s) = 1 | − 0.905·2-s − 0.179·4-s − 0.337·7-s + 1.06·8-s + 0.301·11-s + 1.04·13-s + 0.305·14-s − 0.787·16-s + 1.17·17-s − 1.57·19-s − 0.273·22-s − 1.68·23-s − 0.949·26-s + 0.0606·28-s + 0.0570·29-s + 0.0304·31-s − 0.354·32-s − 1.06·34-s + 1.85·37-s + 1.42·38-s + 1.13·41-s + 1.82·43-s − 0.0542·44-s + 1.52·46-s + 1.72·47-s − 0.886·49-s − 0.188·52-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
| \(L(2)\) |
\(\approx\) |
\(1.068835911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068835911\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.56T + 8T^{2} \) |
| 7 | \( 1 + 6.24T + 343T^{2} \) |
| 13 | \( 1 - 49.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 82.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 5.26T + 2.97e4T^{2} \) |
| 37 | \( 1 - 416.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 513.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 557.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 168.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 786.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 339.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 309.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 798.T + 9.12e5T^{2} \) |
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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
−8.534835576698951568378966764115, −8.036335358098949102577723822730, −7.32649292440202265912358086155, −6.17731322396329652214675722492, −5.78112780741648618080260284891, −4.25544151534436078484081468657, −4.01160124147327670174227973821, −2.59073616851752872733235022870, −1.45701157626238585353517137963, −0.56781221964193608466711584279,
0.56781221964193608466711584279, 1.45701157626238585353517137963, 2.59073616851752872733235022870, 4.01160124147327670174227973821, 4.25544151534436078484081468657, 5.78112780741648618080260284891, 6.17731322396329652214675722492, 7.32649292440202265912358086155, 8.036335358098949102577723822730, 8.534835576698951568378966764115
