| L(s) = 1 | + 4.00·3-s + 11.2·5-s − 34.9·7-s − 10.9·9-s − 33.3·11-s − 20.6·13-s + 45.1·15-s + 93.7·17-s + 105.·19-s − 139.·21-s − 56.0·23-s + 2.48·25-s − 152.·27-s − 7.06·29-s + 263.·31-s − 133.·33-s − 394.·35-s + 333.·37-s − 82.4·39-s − 109.·41-s − 195.·43-s − 123.·45-s − 282.·47-s + 878.·49-s + 375.·51-s + 547.·53-s − 377.·55-s + ⋯ |
| L(s) = 1 | + 0.770·3-s + 1.00·5-s − 1.88·7-s − 0.406·9-s − 0.915·11-s − 0.439·13-s + 0.777·15-s + 1.33·17-s + 1.26·19-s − 1.45·21-s − 0.508·23-s + 0.0198·25-s − 1.08·27-s − 0.0452·29-s + 1.52·31-s − 0.705·33-s − 1.90·35-s + 1.47·37-s − 0.338·39-s − 0.418·41-s − 0.693·43-s − 0.410·45-s − 0.878·47-s + 2.56·49-s + 1.03·51-s + 1.41·53-s − 0.924·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.286657674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.286657674\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 89 | \( 1 - 89T \) |
| good | 3 | \( 1 - 4.00T + 27T^{2} \) |
| 5 | \( 1 - 11.2T + 125T^{2} \) |
| 7 | \( 1 + 34.9T + 343T^{2} \) |
| 11 | \( 1 + 33.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 7.06T + 2.43e4T^{2} \) |
| 31 | \( 1 - 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 109.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 195.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 547.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 2.51T + 2.05e5T^{2} \) |
| 61 | \( 1 - 754.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 131.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 839.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 485.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 443.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 8.16T + 5.71e5T^{2} \) |
| 97 | \( 1 + 1.88e3T + 9.12e5T^{2} \) |
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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
−9.556365829693205511329278016244, −8.378462718158990978097252492996, −7.66528236629142659254899244090, −6.66292876051382680902053717687, −5.84367059476638338069525077915, −5.28420127862275240325750354450, −3.64224263990463088915018747603, −2.93845595318692996044548234740, −2.35383620362100525417933862708, −0.68862105319578762090280406816,
0.68862105319578762090280406816, 2.35383620362100525417933862708, 2.93845595318692996044548234740, 3.64224263990463088915018747603, 5.28420127862275240325750354450, 5.84367059476638338069525077915, 6.66292876051382680902053717687, 7.66528236629142659254899244090, 8.378462718158990978097252492996, 9.556365829693205511329278016244
