| L(s) = 1 | − 4.16·2-s − 0.469·3-s + 9.35·4-s − 20.3·5-s + 1.95·6-s − 2.83·7-s − 5.63·8-s − 26.7·9-s + 84.9·10-s − 11·11-s − 4.39·12-s + 11.8·14-s + 9.57·15-s − 51.3·16-s + 129.·17-s + 111.·18-s + 133.·19-s − 190.·20-s + 1.33·21-s + 45.8·22-s − 40.4·23-s + 2.64·24-s + 290.·25-s + 25.2·27-s − 26.5·28-s − 133.·29-s − 39.8·30-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 0.0903·3-s + 1.16·4-s − 1.82·5-s + 0.133·6-s − 0.153·7-s − 0.248·8-s − 0.991·9-s + 2.68·10-s − 0.301·11-s − 0.105·12-s + 0.225·14-s + 0.164·15-s − 0.802·16-s + 1.84·17-s + 1.46·18-s + 1.61·19-s − 2.13·20-s + 0.0138·21-s + 0.444·22-s − 0.366·23-s + 0.0224·24-s + 2.32·25-s + 0.180·27-s − 0.179·28-s − 0.854·29-s − 0.242·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2724031414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2724031414\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.16T + 8T^{2} \) |
| 3 | \( 1 + 0.469T + 27T^{2} \) |
| 5 | \( 1 + 20.3T + 125T^{2} \) |
| 7 | \( 1 + 2.83T + 343T^{2} \) |
| 17 | \( 1 - 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 40.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 341.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 255.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 475.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 123.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 385.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 99.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 739.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 848.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 380.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 493.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−8.711199189473968082645481063922, −8.015211842217749242459564669421, −7.68831714083356328157700637464, −7.05860879763547598507353271798, −5.76531382878510154538440833639, −4.83869945720256723596999078980, −3.55161520237665805154567869288, −3.00290457920683957598084987430, −1.30963981824671546358533767301, −0.34086641397372680737368897935,
0.34086641397372680737368897935, 1.30963981824671546358533767301, 3.00290457920683957598084987430, 3.55161520237665805154567869288, 4.83869945720256723596999078980, 5.76531382878510154538440833639, 7.05860879763547598507353271798, 7.68831714083356328157700637464, 8.015211842217749242459564669421, 8.711199189473968082645481063922
