L(s) = 1 | − 2-s + 3.40·3-s + 4-s − 2.96·5-s − 3.40·6-s − 1.03·7-s − 8-s + 8.57·9-s + 2.96·10-s + 3.40·12-s + 0.179·13-s + 1.03·14-s − 10.1·15-s + 16-s − 17-s − 8.57·18-s − 4.96·19-s − 2.96·20-s − 3.51·21-s − 5.81·23-s − 3.40·24-s + 3.81·25-s − 0.179·26-s + 18.9·27-s − 1.03·28-s − 7.93·29-s + 10.1·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.96·3-s + 0.5·4-s − 1.32·5-s − 1.38·6-s − 0.390·7-s − 0.353·8-s + 2.85·9-s + 0.939·10-s + 0.982·12-s + 0.0497·13-s + 0.276·14-s − 2.60·15-s + 0.250·16-s − 0.242·17-s − 2.02·18-s − 1.13·19-s − 0.663·20-s − 0.767·21-s − 1.21·23-s − 0.694·24-s + 0.763·25-s − 0.0351·26-s + 3.64·27-s − 0.195·28-s − 1.47·29-s + 1.84·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 3.40T + 3T^{2} \) |
| 5 | \( 1 + 2.96T + 5T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 13 | \( 1 - 0.179T + 13T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 23 | \( 1 + 5.81T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 + 8.14T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 - 8.71T + 43T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 - 6.33T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 2.64T + 89T^{2} \) |
| 97 | \( 1 - 5.24T + 97T^{2} \) |
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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
−8.393523774759303775073048723081, −7.49983160552739747944301870776, −7.13823035646950967182079734102, −6.22309642465228304823822606310, −4.61748018876050639443344522396, −3.85761747912010414472457381565, −3.42350206966008166136334334163, −2.44989244727471719475544343103, −1.64737003955940532972603366467, 0,
1.64737003955940532972603366467, 2.44989244727471719475544343103, 3.42350206966008166136334334163, 3.85761747912010414472457381565, 4.61748018876050639443344522396, 6.22309642465228304823822606310, 7.13823035646950967182079734102, 7.49983160552739747944301870776, 8.393523774759303775073048723081