L(s) = 1 | − 0.615i·2-s − i·3-s + 1.62·4-s + 3.28i·5-s − 0.615·6-s − 2.22i·8-s − 9-s + 2.02·10-s + (−0.0656 − 3.31i)11-s − 1.62i·12-s − 0.142·13-s + 3.28·15-s + 1.87·16-s + 7.26·17-s + 0.615i·18-s + 0.869·19-s + ⋯ |
L(s) = 1 | − 0.435i·2-s − 0.577i·3-s + 0.810·4-s + 1.46i·5-s − 0.251·6-s − 0.788i·8-s − 0.333·9-s + 0.639·10-s + (−0.0197 − 0.999i)11-s − 0.467i·12-s − 0.0395·13-s + 0.848·15-s + 0.467·16-s + 1.76·17-s + 0.145i·18-s + 0.199·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.245491728\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.245491728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.0656 + 3.31i)T \) |
good | 2 | \( 1 + 0.615iT - 2T^{2} \) |
| 5 | \( 1 - 3.28iT - 5T^{2} \) |
| 13 | \( 1 + 0.142T + 13T^{2} \) |
| 17 | \( 1 - 7.26T + 17T^{2} \) |
| 19 | \( 1 - 0.869T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 + 3.70iT - 29T^{2} \) |
| 31 | \( 1 - 4.82iT - 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 - 9.66iT - 43T^{2} \) |
| 47 | \( 1 + 8.76iT - 47T^{2} \) |
| 53 | \( 1 + 6.62T + 53T^{2} \) |
| 59 | \( 1 - 1.54iT - 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 + 7.05iT - 79T^{2} \) |
| 83 | \( 1 + 6.31T + 83T^{2} \) |
| 89 | \( 1 - 7.87iT - 89T^{2} \) |
| 97 | \( 1 - 8.07iT - 97T^{2} \) |
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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
−9.695489970373880215467310474337, −8.150595770220976445922182186346, −7.69867776983694878848159000759, −6.84848973439777075792715819705, −6.21597992975853485793235214401, −5.58539040086481507885727865459, −3.73291994952144785934789539714, −3.07561316341998907285258970519, −2.38578568676956866571694334404, −1.06301171141102809078645476779,
1.16356156927480748589370822572, 2.35783763475709294259235244316, 3.69865182709523768089533870731, 4.67812866156311074578540153174, 5.41931082085881041980964790223, 6.00914399528114512386901367008, 7.26441444900435992949463506975, 7.906851106851282727417100074235, 8.551473554229734916822791348064, 9.693438382137406529891565233792