| L(s) = 1 | + 3·3-s + 19.4·5-s + 17.4·7-s + 9·9-s + 50.9·11-s + 2·13-s + 58.4·15-s − 78.9·17-s + 25.0·19-s + 52.4·21-s − 120.·23-s + 254.·25-s + 27·27-s − 273.·29-s + 29.5·31-s + 152.·33-s + 340.·35-s − 320.·37-s + 6·39-s + 263.·41-s − 388.·43-s + 175.·45-s + 325.·47-s − 37.0·49-s − 236.·51-s − 567.·53-s + 993.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.74·5-s + 0.944·7-s + 0.333·9-s + 1.39·11-s + 0.0426·13-s + 1.00·15-s − 1.12·17-s + 0.302·19-s + 0.545·21-s − 1.09·23-s + 2.03·25-s + 0.192·27-s − 1.75·29-s + 0.171·31-s + 0.806·33-s + 1.64·35-s − 1.42·37-s + 0.0246·39-s + 1.00·41-s − 1.37·43-s + 0.581·45-s + 1.00·47-s − 0.107·49-s − 0.650·51-s − 1.47·53-s + 2.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.654858988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.654858988\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 - 17.4T + 343T^{2} \) |
| 11 | \( 1 - 50.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 273.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 29.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 320.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 263.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 325.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 639.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 176.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 15.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 623.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 305.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 636.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 640.T + 9.12e5T^{2} \) |
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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
−10.83084018112424028192191905248, −9.759276909589729578524797100729, −9.187207540350350340781678401014, −8.380259198152644532687138921626, −7.02687537791179628867064206186, −6.14074839958801291443631185596, −5.10235453668464854722394619668, −3.86062889850492441721457017579, −2.16298215311529397202199265050, −1.54299831947080972240658817244,
1.54299831947080972240658817244, 2.16298215311529397202199265050, 3.86062889850492441721457017579, 5.10235453668464854722394619668, 6.14074839958801291443631185596, 7.02687537791179628867064206186, 8.380259198152644532687138921626, 9.187207540350350340781678401014, 9.759276909589729578524797100729, 10.83084018112424028192191905248
