| L(s) = 1 | + 4.03e3·3-s + 1.63e4·4-s + 7.81e4·5-s − 8.23e5·7-s + 1.14e7·9-s − 3.73e7·11-s + 6.60e7·12-s + 6.52e7·13-s + 3.14e8·15-s + 2.68e8·16-s + 6.26e8·17-s + 1.28e9·20-s − 3.31e9·21-s + 6.10e9·25-s + 2.69e10·27-s − 1.34e10·28-s − 9.77e9·29-s − 1.50e11·33-s − 6.43e10·35-s + 1.87e11·36-s + 2.63e11·39-s − 6.12e11·44-s + 8.95e11·45-s − 7.19e11·47-s + 1.08e12·48-s + 6.78e11·49-s + 2.52e12·51-s + ⋯ |
| L(s) = 1 | + 1.84·3-s + 4-s + 5-s − 7-s + 2.39·9-s − 1.91·11-s + 1.84·12-s + 1.04·13-s + 1.84·15-s + 16-s + 1.52·17-s + 20-s − 1.84·21-s + 25-s + 2.57·27-s − 28-s − 0.566·29-s − 3.53·33-s − 35-s + 2.39·36-s + 1.91·39-s − 1.91·44-s + 2.39·45-s − 1.41·47-s + 1.84·48-s + 49-s + 2.81·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(5.635340991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.635340991\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{7} T \) |
| 7 | \( 1 + p^{7} T \) |
| good | 2 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 3 | \( 1 - 4031 T + p^{14} T^{2} \) |
| 11 | \( 1 + 37379173 T + p^{14} T^{2} \) |
| 13 | \( 1 - 65279611 T + p^{14} T^{2} \) |
| 17 | \( 1 - 626193259 T + p^{14} T^{2} \) |
| 19 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 23 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 29 | \( 1 + 9775649497 T + p^{14} T^{2} \) |
| 31 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 37 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 41 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 43 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 47 | \( 1 + 719081600801 T + p^{14} T^{2} \) |
| 53 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 59 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 61 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 67 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 71 | \( 1 + 1790558995678 T + p^{14} T^{2} \) |
| 73 | \( 1 + 22033597628414 T + p^{14} T^{2} \) |
| 79 | \( 1 + 27088287440917 T + p^{14} T^{2} \) |
| 83 | \( 1 + 33726754263974 T + p^{14} T^{2} \) |
| 89 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 97 | \( 1 + 24587561871581 T + p^{14} T^{2} \) |
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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
−13.36671446924664742961173218605, −12.80547180317572487818995881506, −10.42992026475136935673443571081, −9.764816290900266019797866895817, −8.318992284294079383225843391203, −7.25932772123463287617128120792, −5.77236973797275122858723221866, −3.27275797965958362512429144649, −2.69138256016122048233578304015, −1.50770637997184734306792651613,
1.50770637997184734306792651613, 2.69138256016122048233578304015, 3.27275797965958362512429144649, 5.77236973797275122858723221866, 7.25932772123463287617128120792, 8.318992284294079383225843391203, 9.764816290900266019797866895817, 10.42992026475136935673443571081, 12.80547180317572487818995881506, 13.36671446924664742961173218605
