L(s) = 1 | − 2.13i·2-s − i·3-s − 2.55·4-s − 2.13·6-s + 2.16i·7-s + 1.17i·8-s − 9-s − 2.50·11-s + 2.55i·12-s + 4.33i·13-s + 4.62·14-s − 2.59·16-s − 6.77i·17-s + 2.13i·18-s − 6.83·19-s + ⋯ |
L(s) = 1 | − 1.50i·2-s − 0.577i·3-s − 1.27·4-s − 0.870·6-s + 0.819i·7-s + 0.415i·8-s − 0.333·9-s − 0.756·11-s + 0.736i·12-s + 1.20i·13-s + 1.23·14-s − 0.648·16-s − 1.64i·17-s + 0.502i·18-s − 1.56·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6016902274\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6016902274\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2.13iT - 2T^{2} \) |
| 7 | \( 1 - 2.16iT - 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 - 4.33iT - 13T^{2} \) |
| 17 | \( 1 + 6.77iT - 17T^{2} \) |
| 19 | \( 1 + 6.83T + 19T^{2} \) |
| 23 | \( 1 - 1.67iT - 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 7.97iT - 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 4.25iT - 43T^{2} \) |
| 47 | \( 1 - 4.98iT - 47T^{2} \) |
| 53 | \( 1 - 8.21iT - 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 - 11.5iT - 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 1.11iT - 73T^{2} \) |
| 79 | \( 1 + 7.78T + 79T^{2} \) |
| 83 | \( 1 + 9.87iT - 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 - 15.0iT - 97T^{2} \) |
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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
−9.277017754299327389728315430588, −8.821913200819808681809073089942, −7.79368253481544825437840911739, −6.85269767863287113392112386851, −6.01463016321725039023629310459, −4.87136901419673843201260685245, −4.14976619549863813629264033862, −2.65704855614395004961433314481, −2.52562657560875110062456587205, −1.28751106227011899563800835257,
0.22233631285530341482830569352, 2.34839562406322349716957405540, 3.81366526753807827784344756946, 4.42125959927680940530736577466, 5.45839351158976908439327611265, 5.99340706419077654863258447057, 6.82628492606799924938714939267, 7.77954880795255665857961625477, 8.197589573620851621420950835605, 8.868038499468309202505358786467