| L(s) = 1 | + 1.44·2-s + 0.0881·4-s − 1.35·7-s − 2.76·8-s − 4.85·11-s − 0.198·13-s − 1.96·14-s − 4.16·16-s − 1.13·17-s − 19-s − 7.00·22-s + 2.55·23-s − 0.286·26-s − 0.119·28-s + 10.2·29-s + 2.51·31-s − 0.498·32-s − 1.64·34-s + 0.137·37-s − 1.44·38-s + 11.7·41-s + 7.59·43-s − 0.427·44-s + 3.69·46-s + 2.69·47-s − 5.15·49-s − 0.0174·52-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.0440·4-s − 0.512·7-s − 0.976·8-s − 1.46·11-s − 0.0549·13-s − 0.524·14-s − 1.04·16-s − 0.275·17-s − 0.229·19-s − 1.49·22-s + 0.532·23-s − 0.0561·26-s − 0.0226·28-s + 1.90·29-s + 0.451·31-s − 0.0880·32-s − 0.281·34-s + 0.0225·37-s − 0.234·38-s + 1.83·41-s + 1.15·43-s − 0.0644·44-s + 0.544·46-s + 0.392·47-s − 0.736·49-s − 0.00242·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.991252710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991252710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 + 0.198T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 23 | \( 1 - 2.55T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 - 0.137T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 - 2.69T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 5.82T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 + 8.01T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.77T + 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 + 0.198T + 97T^{2} \) |
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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
−8.341794655262330569395474984256, −7.62506533334567856709254219682, −6.67183755498626942964999695973, −6.05479978319681183259460099830, −5.30117960859128182678004294660, −4.66529267025973110098059542926, −3.96346045918927458421004081203, −2.85056001120812631586795947774, −2.55913061931990186416603032040, −0.65607122455543860387982537649,
0.65607122455543860387982537649, 2.55913061931990186416603032040, 2.85056001120812631586795947774, 3.96346045918927458421004081203, 4.66529267025973110098059542926, 5.30117960859128182678004294660, 6.05479978319681183259460099830, 6.67183755498626942964999695973, 7.62506533334567856709254219682, 8.341794655262330569395474984256
