| L(s) = 1 | + 1.69·2-s − 3-s + 0.859·4-s + 5-s − 1.69·6-s + 4.77·7-s − 1.92·8-s + 9-s + 1.69·10-s − 5.43·11-s − 0.859·12-s + 1.70·13-s + 8.07·14-s − 15-s − 4.98·16-s − 4.01·17-s + 1.69·18-s + 4.39·19-s + 0.859·20-s − 4.77·21-s − 9.18·22-s + 1.92·24-s + 25-s + 2.88·26-s − 27-s + 4.10·28-s − 6.30·29-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.429·4-s + 0.447·5-s − 0.690·6-s + 1.80·7-s − 0.681·8-s + 0.333·9-s + 0.534·10-s − 1.63·11-s − 0.248·12-s + 0.473·13-s + 2.15·14-s − 0.258·15-s − 1.24·16-s − 0.974·17-s + 0.398·18-s + 1.00·19-s + 0.192·20-s − 1.04·21-s − 1.95·22-s + 0.393·24-s + 0.200·25-s + 0.566·26-s − 0.192·27-s + 0.775·28-s − 1.17·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.550036264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.550036264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 + 5.43T + 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 + 4.01T + 17T^{2} \) |
| 19 | \( 1 - 4.39T + 19T^{2} \) |
| 29 | \( 1 + 6.30T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 9.60T + 37T^{2} \) |
| 41 | \( 1 - 4.35T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 - 9.35T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 + 0.902T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 7.15T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 89 | \( 1 - 5.55T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−7.65636594289036963358031442558, −7.10668139152011003574276363496, −6.02013467624773535602245737376, −5.54209230272606407829223183017, −5.04551402332584482035742490562, −4.58130808830912254626035423751, −3.80335258875267622718473581907, −2.63022915417231941388140793885, −2.06159940325041914704213488679, −0.804596468208595886466643692242,
0.804596468208595886466643692242, 2.06159940325041914704213488679, 2.63022915417231941388140793885, 3.80335258875267622718473581907, 4.58130808830912254626035423751, 5.04551402332584482035742490562, 5.54209230272606407829223183017, 6.02013467624773535602245737376, 7.10668139152011003574276363496, 7.65636594289036963358031442558
