| L(s) = 1 | − 2.12·2-s − 3-s + 2.51·4-s + 5-s + 2.12·6-s + 3.70·7-s − 1.09·8-s + 9-s − 2.12·10-s − 4.40·11-s − 2.51·12-s − 1.01·13-s − 7.86·14-s − 15-s − 2.70·16-s + 7.95·17-s − 2.12·18-s − 7.36·19-s + 2.51·20-s − 3.70·21-s + 9.36·22-s − 3.13·23-s + 1.09·24-s + 25-s + 2.14·26-s − 27-s + 9.31·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 0.577·3-s + 1.25·4-s + 0.447·5-s + 0.867·6-s + 1.39·7-s − 0.386·8-s + 0.333·9-s − 0.671·10-s − 1.32·11-s − 0.725·12-s − 0.280·13-s − 2.10·14-s − 0.258·15-s − 0.677·16-s + 1.92·17-s − 0.500·18-s − 1.69·19-s + 0.562·20-s − 0.808·21-s + 1.99·22-s − 0.654·23-s + 0.222·24-s + 0.200·25-s + 0.421·26-s − 0.192·27-s + 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6500101600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6500101600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 + 7.36T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 + 6.96T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 6.99T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 6.72T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 5.52T + 73T^{2} \) |
| 79 | \( 1 + 0.0462T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 4.04T + 97T^{2} \) |
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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
−8.200135353572328586687885164135, −7.61342749242753620281890403626, −6.98675304315678126366278473811, −5.98744420947499491857067607256, −5.22004254128214185468492572127, −4.80547321494564016823833308435, −3.53029400263619853688851606848, −2.02233960119500983372287350925, −1.83987476901751684231163753922, −0.54523790212153944784773445072,
0.54523790212153944784773445072, 1.83987476901751684231163753922, 2.02233960119500983372287350925, 3.53029400263619853688851606848, 4.80547321494564016823833308435, 5.22004254128214185468492572127, 5.98744420947499491857067607256, 6.98675304315678126366278473811, 7.61342749242753620281890403626, 8.200135353572328586687885164135
