| L(s) = 1 | − 2.40·2-s − 1.24·3-s + 3.79·4-s + 3.14·5-s + 2.98·6-s + 7-s − 4.33·8-s − 1.45·9-s − 7.57·10-s − 3.50·11-s − 4.71·12-s + 2.92·13-s − 2.40·14-s − 3.90·15-s + 2.83·16-s + 0.641·17-s + 3.51·18-s + 5.64·19-s + 11.9·20-s − 1.24·21-s + 8.44·22-s − 3.37·23-s + 5.37·24-s + 4.88·25-s − 7.03·26-s + 5.53·27-s + 3.79·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.716·3-s + 1.89·4-s + 1.40·5-s + 1.22·6-s + 0.377·7-s − 1.53·8-s − 0.486·9-s − 2.39·10-s − 1.05·11-s − 1.36·12-s + 0.810·13-s − 0.643·14-s − 1.00·15-s + 0.709·16-s + 0.155·17-s + 0.828·18-s + 1.29·19-s + 2.67·20-s − 0.270·21-s + 1.80·22-s − 0.703·23-s + 1.09·24-s + 0.977·25-s − 1.37·26-s + 1.06·27-s + 0.718·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7992138152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7992138152\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 + 1.24T + 3T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 - 0.641T + 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 - 6.63T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 - 2.46T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 6.16T + 53T^{2} \) |
| 59 | \( 1 - 8.67T + 59T^{2} \) |
| 61 | \( 1 + 1.52T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 0.395T + 79T^{2} \) |
| 83 | \( 1 + 0.703T + 83T^{2} \) |
| 89 | \( 1 - 1.81T + 89T^{2} \) |
| 97 | \( 1 + 0.0756T + 97T^{2} \) |
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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
−8.307316229764159625147486546401, −7.46811465785613764702575910988, −6.81939835317462398693086278577, −5.83761332376507447992487871930, −5.70537221478946024617345132091, −4.80733754312363775210734703892, −3.19164394203989048261153054216, −2.33978841166899807909306541771, −1.55611100372007629017581019581, −0.65465211998505313530493676892,
0.65465211998505313530493676892, 1.55611100372007629017581019581, 2.33978841166899807909306541771, 3.19164394203989048261153054216, 4.80733754312363775210734703892, 5.70537221478946024617345132091, 5.83761332376507447992487871930, 6.81939835317462398693086278577, 7.46811465785613764702575910988, 8.307316229764159625147486546401
