L(s) = 1 | + 1.17·2-s − 1.22·3-s − 0.618·4-s − 1.06·5-s − 1.44·6-s − 3.67·7-s − 3.07·8-s − 1.48·9-s − 1.25·10-s + 11-s + 0.760·12-s − 4.32·14-s + 1.31·15-s − 2.38·16-s + 4.61·17-s − 1.75·18-s − 6.56·19-s + 0.661·20-s + 4.52·21-s + 1.17·22-s + 6.89·23-s + 3.78·24-s − 3.85·25-s + 5.51·27-s + 2.27·28-s − 7.50·29-s + 1.54·30-s + ⋯ |
L(s) = 1 | + 0.831·2-s − 0.709·3-s − 0.309·4-s − 0.478·5-s − 0.589·6-s − 1.39·7-s − 1.08·8-s − 0.496·9-s − 0.397·10-s + 0.301·11-s + 0.219·12-s − 1.15·14-s + 0.339·15-s − 0.595·16-s + 1.11·17-s − 0.412·18-s − 1.50·19-s + 0.147·20-s + 0.986·21-s + 0.250·22-s + 1.43·23-s + 0.772·24-s − 0.771·25-s + 1.06·27-s + 0.429·28-s − 1.39·29-s + 0.282·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7487839335\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7487839335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 17 | \( 1 - 4.61T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 + 7.50T + 29T^{2} \) |
| 31 | \( 1 + 4.02T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 + 2.86T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 7.28T + 59T^{2} \) |
| 61 | \( 1 + 2.10T + 61T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 3.56T + 83T^{2} \) |
| 89 | \( 1 - 3.36T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−9.242883777455611672729479794886, −8.614473202581976067317424540600, −7.45678355463134960128388435445, −6.53162428337687365981035601467, −5.87896861772801098936254664130, −5.34750564549346649036851408154, −4.17783297131565240732947430360, −3.58313066740612074724535609083, −2.68837621750410097977025680084, −0.51067758973398413635862389100,
0.51067758973398413635862389100, 2.68837621750410097977025680084, 3.58313066740612074724535609083, 4.17783297131565240732947430360, 5.34750564549346649036851408154, 5.87896861772801098936254664130, 6.53162428337687365981035601467, 7.45678355463134960128388435445, 8.614473202581976067317424540600, 9.242883777455611672729479794886