| L(s) = 1 | + 0.291·2-s + 3-s − 1.91·4-s + 5-s + 0.291·6-s − 2.46·7-s − 1.14·8-s + 9-s + 0.291·10-s + 5.47·11-s − 1.91·12-s − 5.40·13-s − 0.719·14-s + 15-s + 3.49·16-s − 5.16·17-s + 0.291·18-s + 1.07·19-s − 1.91·20-s − 2.46·21-s + 1.59·22-s + 0.730·23-s − 1.14·24-s + 25-s − 1.57·26-s + 27-s + 4.72·28-s + ⋯ |
| L(s) = 1 | + 0.206·2-s + 0.577·3-s − 0.957·4-s + 0.447·5-s + 0.119·6-s − 0.931·7-s − 0.404·8-s + 0.333·9-s + 0.0923·10-s + 1.65·11-s − 0.552·12-s − 1.49·13-s − 0.192·14-s + 0.258·15-s + 0.873·16-s − 1.25·17-s + 0.0688·18-s + 0.247·19-s − 0.428·20-s − 0.537·21-s + 0.340·22-s + 0.152·23-s − 0.233·24-s + 0.200·25-s − 0.309·26-s + 0.192·27-s + 0.892·28-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
| \(L(1)\) |
\(\approx\) |
\(1.834782767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834782767\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.291T + 2T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 - 0.730T + 23T^{2} \) |
| 29 | \( 1 - 0.408T + 29T^{2} \) |
| 31 | \( 1 - 0.963T + 31T^{2} \) |
| 37 | \( 1 + 0.284T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 0.188T + 43T^{2} \) |
| 47 | \( 1 - 2.69T + 47T^{2} \) |
| 53 | \( 1 + 8.70T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 0.355T + 61T^{2} \) |
| 67 | \( 1 - 2.62T + 67T^{2} \) |
| 71 | \( 1 - 9.07T + 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.251371891908502012344759538903, −7.25523428338163516245513876219, −6.65561511425363969190659047381, −6.03688404114973384027721132986, −5.01486803114928432335830656649, −4.38639410869780075185087467356, −3.70059021317082164329809873474, −2.89905937168414230260688232660, −1.98024894995081678452274141873, −0.66594733043087471809136892895,
0.66594733043087471809136892895, 1.98024894995081678452274141873, 2.89905937168414230260688232660, 3.70059021317082164329809873474, 4.38639410869780075185087467356, 5.01486803114928432335830656649, 6.03688404114973384027721132986, 6.65561511425363969190659047381, 7.25523428338163516245513876219, 8.251371891908502012344759538903
