| L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s + 4·7-s − 8-s + 9-s − 4·10-s − 11-s + 12-s + 6·13-s − 4·14-s + 4·15-s + 16-s − 6·17-s − 18-s + 4·20-s + 4·21-s + 22-s − 4·23-s − 24-s + 11·25-s − 6·26-s + 27-s + 4·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s + 0.288·12-s + 1.66·13-s − 1.06·14-s + 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.894·20-s + 0.872·21-s + 0.213·22-s − 0.834·23-s − 0.204·24-s + 11/5·25-s − 1.17·26-s + 0.192·27-s + 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.094734690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.094734690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.504813622403158755999790179386, −7.983539313733492512983175922018, −7.09198953830004083511910783494, −6.14584505279045050546879567889, −5.72584957774270213764923032130, −4.75766149710626322437813506935, −3.77875930935357350291687021724, −2.38081091892774572566155165472, −1.95477332916121725533922654784, −1.22922377685872579326064454995,
1.22922377685872579326064454995, 1.95477332916121725533922654784, 2.38081091892774572566155165472, 3.77875930935357350291687021724, 4.75766149710626322437813506935, 5.72584957774270213764923032130, 6.14584505279045050546879567889, 7.09198953830004083511910783494, 7.983539313733492512983175922018, 8.504813622403158755999790179386
