| L(s) = 1 | − 2.84e3·2-s + 5.90e4·3-s + 6.00e6·4-s − 1.68e8·6-s − 1.14e9·7-s − 1.11e10·8-s + 3.48e9·9-s − 1.26e11·11-s + 3.54e11·12-s + 5.22e11·13-s + 3.24e12·14-s + 1.90e13·16-s − 5.12e12·17-s − 9.92e12·18-s + 4.43e13·19-s − 6.73e13·21-s + 3.61e14·22-s − 7.57e13·23-s − 6.55e14·24-s − 1.48e15·26-s + 2.05e14·27-s − 6.84e15·28-s + 2.13e15·29-s − 4.16e15·31-s − 3.08e16·32-s − 7.49e15·33-s + 1.45e16·34-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 0.577·3-s + 2.86·4-s − 1.13·6-s − 1.52·7-s − 3.65·8-s + 0.333·9-s − 1.47·11-s + 1.65·12-s + 1.05·13-s + 2.99·14-s + 4.32·16-s − 0.617·17-s − 0.655·18-s + 1.65·19-s − 0.881·21-s + 2.89·22-s − 0.381·23-s − 2.11·24-s − 2.06·26-s + 0.192·27-s − 4.36·28-s + 0.944·29-s − 0.913·31-s − 4.84·32-s − 0.851·33-s + 1.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.4630109728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4630109728\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.84e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 1.14e9T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.26e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 5.22e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 5.12e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.43e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 7.57e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.13e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 4.16e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.63e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 2.05e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.18e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.51e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 5.57e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.75e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.15e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 5.26e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.86e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 8.35e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 7.86e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 3.57e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.60e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.39e21T + 5.27e41T^{2} \) |
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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
−10.18067725991655338625633910910, −9.561264002797974346270474841042, −8.636327097980075780851639438873, −7.72521369006353283735259419587, −6.81599492140050295759208596823, −5.80823755906910219837096187609, −3.26926067744868852604662385370, −2.75998167070709957821139022668, −1.50559314544043013522119746209, −0.36992845583697181659096923694,
0.36992845583697181659096923694, 1.50559314544043013522119746209, 2.75998167070709957821139022668, 3.26926067744868852604662385370, 5.80823755906910219837096187609, 6.81599492140050295759208596823, 7.72521369006353283735259419587, 8.636327097980075780851639438873, 9.561264002797974346270474841042, 10.18067725991655338625633910910
