| L(s) = 1 | + 3.88·2-s − 1.79·3-s + 11.0·4-s − 6.96·6-s + 7·7-s + 27.5·8-s − 5.78·9-s − 19.8·12-s − 0.743·13-s + 27.1·14-s + 62.5·16-s − 30.1·17-s − 22.4·18-s + 37.2·19-s − 12.5·21-s − 40.5·23-s − 49.3·24-s + 25·25-s − 2.88·26-s + 26.5·27-s + 77.5·28-s + 132.·32-s − 117.·34-s − 64.0·36-s − 61.3·37-s + 144.·38-s + 1.33·39-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.598·3-s + 2.77·4-s − 1.16·6-s + 7-s + 3.43·8-s − 0.642·9-s − 1.65·12-s − 0.0571·13-s + 1.94·14-s + 3.90·16-s − 1.77·17-s − 1.24·18-s + 1.96·19-s − 0.598·21-s − 1.76·23-s − 2.05·24-s + 25-s − 0.111·26-s + 0.982·27-s + 2.77·28-s + 4.14·32-s − 3.44·34-s − 1.77·36-s − 1.65·37-s + 3.80·38-s + 0.0341·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.499912509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.499912509\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 - 3.88T + 4T^{2} \) |
| 3 | \( 1 + 1.79T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 0.743T + 169T^{2} \) |
| 17 | \( 1 + 30.1T + 289T^{2} \) |
| 19 | \( 1 - 37.2T + 361T^{2} \) |
| 23 | \( 1 + 40.5T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 61.3T + 1.36e3T^{2} \) |
| 43 | \( 1 + 8.84T + 1.84e3T^{2} \) |
| 47 | \( 1 + 93.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 36.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 13.3T + 9.40e3T^{2} \) |
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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
−11.62477358086435173490321807426, −11.32637525308890480616334653925, −10.31448502196522168016343725552, −8.414571481822999774818192853151, −7.20403565854595623865802303397, −6.26868768585645916483435625589, −5.25025868898470559821727610661, −4.69707453353672683798745677959, −3.32472433435917167863269346200, −1.92207827360599397437698187955,
1.92207827360599397437698187955, 3.32472433435917167863269346200, 4.69707453353672683798745677959, 5.25025868898470559821727610661, 6.26868768585645916483435625589, 7.20403565854595623865802303397, 8.414571481822999774818192853151, 10.31448502196522168016343725552, 11.32637525308890480616334653925, 11.62477358086435173490321807426
