| L(s) = 1 | + 0.424·3-s + 10.4·5-s − 34.5·7-s − 26.8·9-s − 11·11-s − 1.15·13-s + 4.42·15-s − 60.3·17-s + 92.4·19-s − 14.6·21-s − 53.7·23-s − 16.3·25-s − 22.8·27-s − 275.·29-s − 195.·31-s − 4.66·33-s − 360.·35-s + 283.·37-s − 0.488·39-s − 70.3·41-s − 96.9·43-s − 279.·45-s + 510.·47-s + 850.·49-s − 25.5·51-s − 298.·53-s − 114.·55-s + ⋯ |
| L(s) = 1 | + 0.0816·3-s + 0.932·5-s − 1.86·7-s − 0.993·9-s − 0.301·11-s − 0.0245·13-s + 0.0761·15-s − 0.860·17-s + 1.11·19-s − 0.152·21-s − 0.487·23-s − 0.130·25-s − 0.162·27-s − 1.76·29-s − 1.13·31-s − 0.0246·33-s − 1.73·35-s + 1.26·37-s − 0.00200·39-s − 0.268·41-s − 0.343·43-s − 0.926·45-s + 1.58·47-s + 2.47·49-s − 0.0702·51-s − 0.773·53-s − 0.281·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 0.424T + 27T^{2} \) |
| 5 | \( 1 - 10.4T + 125T^{2} \) |
| 7 | \( 1 + 34.5T + 343T^{2} \) |
| 13 | \( 1 + 1.15T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 92.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 275.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 70.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 96.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 510.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 298.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 261.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 418.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 304.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 641.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 182.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 806.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 926.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.80e3T + 9.12e5T^{2} \) |
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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
−11.80072996140007928203738048810, −10.60732924059941439256303126354, −9.512122545396612462615701465218, −9.132274925974300993556158061283, −7.46020362093846161507691730789, −6.20465825363728412460177000427, −5.58686854697651785974993535960, −3.55809675476593822906278627100, −2.40671834972962079853808266450, 0,
2.40671834972962079853808266450, 3.55809675476593822906278627100, 5.58686854697651785974993535960, 6.20465825363728412460177000427, 7.46020362093846161507691730789, 9.132274925974300993556158061283, 9.512122545396612462615701465218, 10.60732924059941439256303126354, 11.80072996140007928203738048810
