| L(s) = 1 | − 1.53i·2-s + i·3-s − 0.364·4-s + 1.53·6-s + 1.68i·7-s − 2.51i·8-s − 9-s − 2.97·11-s − 0.364i·12-s − 0.232i·13-s + 2.59·14-s − 4.59·16-s + 7.45i·17-s + 1.53i·18-s − 0.753·19-s + ⋯ |
| L(s) = 1 | − 1.08i·2-s + 0.577i·3-s − 0.182·4-s + 0.627·6-s + 0.637i·7-s − 0.889i·8-s − 0.333·9-s − 0.897·11-s − 0.105i·12-s − 0.0645i·13-s + 0.692·14-s − 1.14·16-s + 1.80i·17-s + 0.362i·18-s − 0.172·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7026117483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7026117483\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.53iT - 2T^{2} \) |
| 7 | \( 1 - 1.68iT - 7T^{2} \) |
| 11 | \( 1 + 2.97T + 11T^{2} \) |
| 13 | \( 1 + 0.232iT - 13T^{2} \) |
| 17 | \( 1 - 7.45iT - 17T^{2} \) |
| 19 | \( 1 + 0.753T + 19T^{2} \) |
| 23 | \( 1 + 0.872iT - 23T^{2} \) |
| 29 | \( 1 + 6.87T + 29T^{2} \) |
| 31 | \( 1 + 9.81T + 31T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 - 5.27iT - 43T^{2} \) |
| 47 | \( 1 + 8.56iT - 47T^{2} \) |
| 53 | \( 1 + 5.97iT - 53T^{2} \) |
| 59 | \( 1 - 3.85T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 - 1.79iT - 67T^{2} \) |
| 71 | \( 1 + 4.37T + 71T^{2} \) |
| 73 | \( 1 - 15.0iT - 73T^{2} \) |
| 79 | \( 1 + 7.37T + 79T^{2} \) |
| 83 | \( 1 + 4.34iT - 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.47iT - 97T^{2} \) |
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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
−9.692935328283473696433025384770, −8.813141793429107860070660940932, −8.133808506787857937562900590671, −7.07403183758328225674382181896, −6.01568273408854165207441549096, −5.34311503320807342420373426916, −4.16847927458703746206682019238, −3.47769162408851830981928585914, −2.51205577883179882120818359832, −1.65399119437399446748084905264,
0.23142609552918511639157282568, 1.97087407489399654402233162990, 2.95186352380693288215959790737, 4.29930505271053676531334569381, 5.41048955421266448346848360835, 5.76610765218991582930311789356, 7.01067967774898110810115182258, 7.38078355496963209883873688436, 7.76152238564508317243316089500, 8.911338136026579555683412831303
