| L(s) = 1 | − 2-s − 0.343·3-s + 4-s + 2.06·5-s + 0.343·6-s − 1.08·7-s − 8-s − 2.88·9-s − 2.06·10-s − 6.18·11-s − 0.343·12-s + 1.08·14-s − 0.709·15-s + 16-s − 6.94·17-s + 2.88·18-s − 19-s + 2.06·20-s + 0.374·21-s + 6.18·22-s + 7.55·23-s + 0.343·24-s − 0.740·25-s + 2.02·27-s − 1.08·28-s + 4.02·29-s + 0.709·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.198·3-s + 0.5·4-s + 0.922·5-s + 0.140·6-s − 0.411·7-s − 0.353·8-s − 0.960·9-s − 0.652·10-s − 1.86·11-s − 0.0992·12-s + 0.290·14-s − 0.183·15-s + 0.250·16-s − 1.68·17-s + 0.679·18-s − 0.229·19-s + 0.461·20-s + 0.0816·21-s + 1.31·22-s + 1.57·23-s + 0.0701·24-s − 0.148·25-s + 0.389·27-s − 0.205·28-s + 0.747·29-s + 0.129·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6493673499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6493673499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.343T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 + 6.18T + 11T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 4.82T + 59T^{2} \) |
| 61 | \( 1 - 7.45T + 61T^{2} \) |
| 67 | \( 1 - 5.83T + 67T^{2} \) |
| 71 | \( 1 + 5.90T + 71T^{2} \) |
| 73 | \( 1 - 2.20T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 6.19T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.353435522098665608178454618228, −7.22141495129690405877509495165, −6.68321681269313004442192326965, −5.96122871590342906576280634192, −5.29422161450026363053999938620, −4.70514822884947259432526143250, −3.17790479528105015815931164114, −2.61814077028477735440087496793, −1.93592738451580045795449363492, −0.43685598825850214943889607098,
0.43685598825850214943889607098, 1.93592738451580045795449363492, 2.61814077028477735440087496793, 3.17790479528105015815931164114, 4.70514822884947259432526143250, 5.29422161450026363053999938620, 5.96122871590342906576280634192, 6.68321681269313004442192326965, 7.22141495129690405877509495165, 8.353435522098665608178454618228
