L(s) = 1 | − 516·3-s − 1.05e4·5-s + 4.93e4·7-s + 8.91e4·9-s − 3.09e5·11-s − 1.72e6·13-s + 5.43e6·15-s − 2.27e6·17-s + 4.55e6·19-s − 2.54e7·21-s − 7.28e6·23-s + 6.20e7·25-s + 4.54e7·27-s − 6.90e7·29-s − 1.41e8·31-s + 1.59e8·33-s − 5.19e8·35-s + 7.11e8·37-s + 8.89e8·39-s − 1.22e9·41-s − 3.36e7·43-s − 9.38e8·45-s + 1.23e8·47-s + 4.53e8·49-s + 1.17e9·51-s + 1.10e9·53-s + 3.25e9·55-s + ⋯ |
L(s) = 1 | − 1.22·3-s − 1.50·5-s + 1.10·7-s + 0.503·9-s − 0.579·11-s − 1.28·13-s + 1.84·15-s − 0.389·17-s + 0.421·19-s − 1.35·21-s − 0.235·23-s + 1.27·25-s + 0.609·27-s − 0.625·29-s − 0.889·31-s + 0.710·33-s − 1.67·35-s + 1.68·37-s + 1.57·39-s − 1.65·41-s − 0.0348·43-s − 0.758·45-s + 0.0783·47-s + 0.229·49-s + 0.477·51-s + 0.363·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 172 p T + p^{11} T^{2} \) |
| 5 | \( 1 + 2106 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 49304 T + p^{11} T^{2} \) |
| 11 | \( 1 + 309420 T + p^{11} T^{2} \) |
| 13 | \( 1 + 1723594 T + p^{11} T^{2} \) |
| 17 | \( 1 + 2279502 T + p^{11} T^{2} \) |
| 19 | \( 1 - 4550444 T + p^{11} T^{2} \) |
| 23 | \( 1 + 7282872 T + p^{11} T^{2} \) |
| 29 | \( 1 + 69040026 T + p^{11} T^{2} \) |
| 31 | \( 1 + 141740704 T + p^{11} T^{2} \) |
| 37 | \( 1 - 711366974 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1225262214 T + p^{11} T^{2} \) |
| 43 | \( 1 + 781540 p T + p^{11} T^{2} \) |
| 47 | \( 1 - 123214608 T + p^{11} T^{2} \) |
| 53 | \( 1 - 1106121582 T + p^{11} T^{2} \) |
| 59 | \( 1 + 9062779932 T + p^{11} T^{2} \) |
| 61 | \( 1 + 3854150458 T + p^{11} T^{2} \) |
| 67 | \( 1 + 15313764676 T + p^{11} T^{2} \) |
| 71 | \( 1 - 20619626328 T + p^{11} T^{2} \) |
| 73 | \( 1 + 2063718694 T + p^{11} T^{2} \) |
| 79 | \( 1 - 13689871472 T + p^{11} T^{2} \) |
| 83 | \( 1 - 65570428908 T + p^{11} T^{2} \) |
| 89 | \( 1 + 29715508854 T + p^{11} T^{2} \) |
| 97 | \( 1 + 23439626206 T + p^{11} T^{2} \) |
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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
−22.06109866433068452206172557423, −20.04950916627822221709274531585, −18.20947205296923602811462763920, −16.69000482504312290024239118396, −15.06487205383862800159819581966, −12.08409413443952132970218524204, −11.07944925540005663929620431223, −7.67375813661486717198729788923, −4.85967772325160364098586662825, 0,
4.85967772325160364098586662825, 7.67375813661486717198729788923, 11.07944925540005663929620431223, 12.08409413443952132970218524204, 15.06487205383862800159819581966, 16.69000482504312290024239118396, 18.20947205296923602811462763920, 20.04950916627822221709274531585, 22.06109866433068452206172557423