| L(s) = 1 | − 2.53·2-s + 1.72·3-s + 4.40·4-s − 1.30·5-s − 4.36·6-s + 7-s − 6.09·8-s − 0.0235·9-s + 3.29·10-s − 0.721·11-s + 7.60·12-s + 0.841·13-s − 2.53·14-s − 2.24·15-s + 6.61·16-s + 7.21·17-s + 0.0596·18-s − 3.82·19-s − 5.74·20-s + 1.72·21-s + 1.82·22-s + 0.353·23-s − 10.5·24-s − 3.30·25-s − 2.13·26-s − 5.21·27-s + 4.40·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.996·3-s + 2.20·4-s − 0.582·5-s − 1.78·6-s + 0.377·7-s − 2.15·8-s − 0.00785·9-s + 1.04·10-s − 0.217·11-s + 2.19·12-s + 0.233·13-s − 0.676·14-s − 0.580·15-s + 1.65·16-s + 1.75·17-s + 0.0140·18-s − 0.876·19-s − 1.28·20-s + 0.376·21-s + 0.389·22-s + 0.0736·23-s − 2.14·24-s − 0.660·25-s − 0.417·26-s − 1.00·27-s + 0.832·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019230146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019230146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 859 | \( 1 + T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 11 | \( 1 + 0.721T + 11T^{2} \) |
| 13 | \( 1 - 0.841T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 - 0.353T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 - 5.64T + 31T^{2} \) |
| 37 | \( 1 - 0.891T + 37T^{2} \) |
| 41 | \( 1 - 5.49T + 41T^{2} \) |
| 43 | \( 1 + 8.06T + 43T^{2} \) |
| 47 | \( 1 + 3.08T + 47T^{2} \) |
| 53 | \( 1 - 9.23T + 53T^{2} \) |
| 59 | \( 1 - 2.08T + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 - 4.81T + 67T^{2} \) |
| 71 | \( 1 - 1.37T + 71T^{2} \) |
| 73 | \( 1 - 6.46T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 - 2.89T + 83T^{2} \) |
| 89 | \( 1 + 5.28T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.238846155383625052732600415265, −7.84258097338706188376739424481, −7.12164559969455042495836507661, −6.28959540031380013083897101105, −5.41034941962122962237562534584, −4.15823347980466906799414182048, −3.26309742144158716853039681522, −2.55864243355298550709349774394, −1.68807996976652038746540602710, −0.66762521015439969907396692028,
0.66762521015439969907396692028, 1.68807996976652038746540602710, 2.55864243355298550709349774394, 3.26309742144158716853039681522, 4.15823347980466906799414182048, 5.41034941962122962237562534584, 6.28959540031380013083897101105, 7.12164559969455042495836507661, 7.84258097338706188376739424481, 8.238846155383625052732600415265
