| L(s) = 1 | + 2-s + 3-s + 4-s − 3.51·5-s + 6-s + 8-s + 9-s − 3.51·10-s + 3.45·11-s + 12-s + 13-s − 3.51·15-s + 16-s − 1.45·17-s + 18-s − 3.51·19-s − 3.51·20-s + 3.45·22-s + 5.51·23-s + 24-s + 7.34·25-s + 26-s + 27-s − 0.869·29-s − 3.51·30-s + 4.96·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.57·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.11·10-s + 1.04·11-s + 0.288·12-s + 0.277·13-s − 0.907·15-s + 0.250·16-s − 0.352·17-s + 0.235·18-s − 0.805·19-s − 0.785·20-s + 0.736·22-s + 1.14·23-s + 0.204·24-s + 1.46·25-s + 0.196·26-s + 0.192·27-s − 0.161·29-s − 0.641·30-s + 0.892·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.978499639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.978499639\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.51T + 5T^{2} \) |
| 11 | \( 1 - 3.45T + 11T^{2} \) |
| 17 | \( 1 + 1.45T + 17T^{2} \) |
| 19 | \( 1 + 3.51T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 + 0.869T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 - 3.55T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 + 2.72T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.96T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 0.486T + 73T^{2} \) |
| 79 | \( 1 + 8.62T + 79T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 - 2.08T + 97T^{2} \) |
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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
−8.512198484401419580142140877795, −7.73458949160604605443764297798, −6.86932311850730409196035929673, −6.60349872942954370879200054069, −5.27321299585697701137006245218, −4.41831847213850327578222551332, −3.85870836212939102060008772723, −3.31716120465234644502125048628, −2.23713187825935637898895269363, −0.892701342458322731136182492659,
0.892701342458322731136182492659, 2.23713187825935637898895269363, 3.31716120465234644502125048628, 3.85870836212939102060008772723, 4.41831847213850327578222551332, 5.27321299585697701137006245218, 6.60349872942954370879200054069, 6.86932311850730409196035929673, 7.73458949160604605443764297798, 8.512198484401419580142140877795
