| L(s) = 1 | − 22.6·2-s − 201.·3-s + 1.69·4-s − 1.57e3·5-s + 4.56e3·6-s + 8.23e3·7-s + 1.15e4·8-s + 2.08e4·9-s + 3.56e4·10-s − 8.15e4·11-s − 340.·12-s + 1.79e5·13-s − 1.86e5·14-s + 3.16e5·15-s − 2.63e5·16-s + 3.07e5·17-s − 4.72e5·18-s + 1.35e4·19-s − 2.66e3·20-s − 1.65e6·21-s + 1.84e6·22-s − 1.39e6·23-s − 2.32e6·24-s + 5.26e5·25-s − 4.07e6·26-s − 2.31e5·27-s + 1.39e4·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 1.43·3-s + 0.00330·4-s − 1.12·5-s + 1.43·6-s + 1.29·7-s + 0.998·8-s + 1.05·9-s + 1.12·10-s − 1.67·11-s − 0.00474·12-s + 1.74·13-s − 1.29·14-s + 1.61·15-s − 1.00·16-s + 0.892·17-s − 1.06·18-s + 0.0239·19-s − 0.00372·20-s − 1.86·21-s + 1.68·22-s − 1.04·23-s − 1.43·24-s + 0.269·25-s − 1.75·26-s − 0.0839·27-s + 0.00428·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 1.87e6T \) |
| good | 2 | \( 1 + 22.6T + 512T^{2} \) |
| 3 | \( 1 + 201.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.57e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.23e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.15e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.79e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.07e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.35e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.39e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.77e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.98e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 1.25e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.67e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.50e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.50e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.86e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 4.92e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 8.38e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.27e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.06e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.92e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.23e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.44e8T + 7.60e17T^{2} \) |
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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
−13.63703997662973875106078284476, −12.02700877591522789306603338190, −11.04667876938706356867411612919, −10.45603392082181441782709815222, −8.294374506507821197493861127272, −7.70510192202934123886169752517, −5.60395196130659795901641707433, −4.34826229053114272615957026082, −1.14740040801441444243765740808, 0,
1.14740040801441444243765740808, 4.34826229053114272615957026082, 5.60395196130659795901641707433, 7.70510192202934123886169752517, 8.294374506507821197493861127272, 10.45603392082181441782709815222, 11.04667876938706356867411612919, 12.02700877591522789306603338190, 13.63703997662973875106078284476
