L(s) = 1 | − 15.3·5-s + 7·7-s − 7.38·11-s − 67.5·13-s + 138.·17-s + 116.·19-s + 133.·23-s + 111.·25-s + 120·29-s − 295.·31-s − 107.·35-s − 337.·37-s + 409.·41-s − 346.·43-s − 345.·47-s + 49·49-s − 381.·53-s + 113.·55-s − 438.·59-s + 707.·61-s + 1.03e3·65-s − 264.·67-s − 391.·71-s − 237.·73-s − 51.6·77-s + 1.14e3·79-s + 49.1·83-s + ⋯ |
L(s) = 1 | − 1.37·5-s + 0.377·7-s − 0.202·11-s − 1.44·13-s + 1.97·17-s + 1.40·19-s + 1.21·23-s + 0.892·25-s + 0.768·29-s − 1.71·31-s − 0.519·35-s − 1.50·37-s + 1.55·41-s − 1.22·43-s − 1.07·47-s + 0.142·49-s − 0.990·53-s + 0.278·55-s − 0.966·59-s + 1.48·61-s + 1.98·65-s − 0.483·67-s − 0.655·71-s − 0.380·73-s − 0.0764·77-s + 1.62·79-s + 0.0649·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
good | 5 | \( 1 + 15.3T + 125T^{2} \) |
| 11 | \( 1 + 7.38T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 138.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 133.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120T + 2.43e4T^{2} \) |
| 31 | \( 1 + 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 409.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 381.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 438.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 707.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 264.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 391.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 49.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 333.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 397.T + 9.12e5T^{2} \) |
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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
−9.203025440156625895459642709955, −8.028146646084046707938267331122, −7.59214332077242201686721227922, −7.00052924525136150026917798370, −5.35511213706196226403091359465, −4.93141306883960572616394227371, −3.63257720088222568193421183236, −2.95733698825324086143225739488, −1.25952188797943989360061972543, 0,
1.25952188797943989360061972543, 2.95733698825324086143225739488, 3.63257720088222568193421183236, 4.93141306883960572616394227371, 5.35511213706196226403091359465, 7.00052924525136150026917798370, 7.59214332077242201686721227922, 8.028146646084046707938267331122, 9.203025440156625895459642709955