L(s) = 1 | − 2-s − 0.848·3-s + 4-s + 4.13·5-s + 0.848·6-s + 1.58·7-s − 8-s − 2.28·9-s − 4.13·10-s + 5.98·11-s − 0.848·12-s + 0.0719·13-s − 1.58·14-s − 3.50·15-s + 16-s + 4.53·17-s + 2.28·18-s + 5.29·19-s + 4.13·20-s − 1.34·21-s − 5.98·22-s + 2.66·23-s + 0.848·24-s + 12.0·25-s − 0.0719·26-s + 4.47·27-s + 1.58·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.489·3-s + 0.5·4-s + 1.84·5-s + 0.346·6-s + 0.599·7-s − 0.353·8-s − 0.760·9-s − 1.30·10-s + 1.80·11-s − 0.244·12-s + 0.0199·13-s − 0.423·14-s − 0.905·15-s + 0.250·16-s + 1.10·17-s + 0.537·18-s + 1.21·19-s + 0.924·20-s − 0.293·21-s − 1.27·22-s + 0.554·23-s + 0.173·24-s + 2.41·25-s − 0.0141·26-s + 0.861·27-s + 0.299·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.419042226\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.419042226\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
good | 3 | \( 1 + 0.848T + 3T^{2} \) |
| 5 | \( 1 - 4.13T + 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 - 5.98T + 11T^{2} \) |
| 13 | \( 1 - 0.0719T + 13T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 - 0.00414T + 47T^{2} \) |
| 53 | \( 1 + 0.527T + 53T^{2} \) |
| 59 | \( 1 + 8.17T + 59T^{2} \) |
| 61 | \( 1 + 2.13T + 61T^{2} \) |
| 67 | \( 1 + 8.75T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 4.79T + 83T^{2} \) |
| 89 | \( 1 + 6.54T + 89T^{2} \) |
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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
−8.257743717051043327157392257219, −7.27545909323446500037191640567, −6.49461138031467567350709066451, −6.07728935368278792503931369281, −5.40786513356803797745061943703, −4.78285294659031939588563720124, −3.36371092470164203294357069412, −2.60041818720089207003016992365, −1.37914273276224709743022514831, −1.16860418762944799044130167872,
1.16860418762944799044130167872, 1.37914273276224709743022514831, 2.60041818720089207003016992365, 3.36371092470164203294357069412, 4.78285294659031939588563720124, 5.40786513356803797745061943703, 6.07728935368278792503931369281, 6.49461138031467567350709066451, 7.27545909323446500037191640567, 8.257743717051043327157392257219