L(s) = 1 | − i·3-s + (−1 − 2i)5-s − 9-s − 4·11-s + (−2 + i)15-s + 4i·17-s + 4i·23-s + (−3 + 4i)25-s + i·27-s + 6·29-s − 4·31-s + 4i·33-s + 8i·37-s + 10·41-s + 4i·43-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.447 − 0.894i)5-s − 0.333·9-s − 1.20·11-s + (−0.516 + 0.258i)15-s + 0.970i·17-s + 0.834i·23-s + (−0.600 + 0.800i)25-s + 0.192i·27-s + 1.11·29-s − 0.718·31-s + 0.696i·33-s + 1.31i·37-s + 1.56·41-s + 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.017452045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.017452045\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.515115661106472199158030756414, −8.111349951585440869201714362962, −7.49930856919394887975895744223, −6.57169978731651954784374164866, −5.64312235629762590557318185782, −5.05627106343126384645776604662, −4.12099934700969364825161472499, −3.13662821620282539175315932176, −2.02860675558636914170257481991, −0.949879953779989810366524534503,
0.38912957418605204233968045201, 2.43433062179563664002280813199, 2.91083144254592597922393623083, 3.96588714301193085480673723765, 4.73641686312773950260903337717, 5.59213847209174319582998412860, 6.41051605657725976624063571491, 7.37719524432752333665482498017, 7.77839136969694624616842462399, 8.738659136578894848488510340048