| L(s) = 1 | − i·2-s + 3.40·3-s − 4-s − 3.40i·6-s − 2.06·7-s + i·8-s + 8.58·9-s + 3.77·11-s − 3.40·12-s + 2.88i·13-s + 2.06i·14-s + 16-s + 5.80i·17-s − 8.58i·18-s − 0.157i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.96·3-s − 0.5·4-s − 1.38i·6-s − 0.779·7-s + 0.353i·8-s + 2.86·9-s + 1.13·11-s − 0.982·12-s + 0.799i·13-s + 0.551i·14-s + 0.250·16-s + 1.40i·17-s − 2.02i·18-s − 0.0360i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.446828467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.446828467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + (-3.72 + 4.80i)T \) |
| good | 3 | \( 1 - 3.40T + 3T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 - 2.88iT - 13T^{2} \) |
| 17 | \( 1 - 5.80iT - 17T^{2} \) |
| 19 | \( 1 + 0.157iT - 19T^{2} \) |
| 23 | \( 1 + 5.41iT - 23T^{2} \) |
| 29 | \( 1 + 4.29iT - 29T^{2} \) |
| 31 | \( 1 + 0.425iT - 31T^{2} \) |
| 41 | \( 1 - 0.923T + 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 0.676T + 47T^{2} \) |
| 53 | \( 1 - 9.87T + 53T^{2} \) |
| 59 | \( 1 + 8.47iT - 59T^{2} \) |
| 61 | \( 1 + 1.23iT - 61T^{2} \) |
| 67 | \( 1 + 6.45T + 67T^{2} \) |
| 71 | \( 1 + 3.28T + 71T^{2} \) |
| 73 | \( 1 + 0.980T + 73T^{2} \) |
| 79 | \( 1 - 8.04iT - 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 7.65iT - 89T^{2} \) |
| 97 | \( 1 - 13.9iT - 97T^{2} \) |
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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
−9.214942365338988074934283032989, −8.578190732294933804327831472986, −7.947463482047548613170900461934, −6.88137248876202654714428563394, −6.22563038598060841387888889033, −4.29404170921769506022173626035, −4.06645733088747439881214741854, −3.13850352236550397643404393148, −2.27735396748215297371463974358, −1.38651683944005690229108413769,
1.24736442191049282842513269470, 2.70306915766960357429838551257, 3.41016719586176093340388928870, 4.11430817055557634414593449107, 5.23051063679846386766424344572, 6.48184124503037143187959199684, 7.24715112398254037442455620590, 7.62965754213292730756557237870, 8.766910823871792787409078784355, 9.006977640754308685891294282809
