L(s) = 1 | + (−1.71 − 0.242i)3-s + 3.08·7-s + (2.88 + 0.831i)9-s − 2.54i·11-s + 5.06·13-s + 0.214·17-s + 2.60·19-s + (−5.29 − 0.749i)21-s + 4.47i·23-s + (−4.74 − 2.12i)27-s + 7.86·29-s − 4.58i·31-s + (−0.617 + 4.36i)33-s − 7.67·37-s + (−8.68 − 1.22i)39-s + ⋯ |
L(s) = 1 | + (−0.990 − 0.139i)3-s + 1.16·7-s + (0.960 + 0.277i)9-s − 0.767i·11-s + 1.40·13-s + 0.0519·17-s + 0.598·19-s + (−1.15 − 0.163i)21-s + 0.933i·23-s + (−0.912 − 0.408i)27-s + 1.46·29-s − 0.823i·31-s + (−0.107 + 0.760i)33-s − 1.26·37-s + (−1.39 − 0.196i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.681154407\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.681154407\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.71 + 0.242i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 + 2.54iT - 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 - 0.214T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 4.58iT - 31T^{2} \) |
| 37 | \( 1 + 7.67T + 37T^{2} \) |
| 41 | \( 1 + 9.26iT - 41T^{2} \) |
| 43 | \( 1 - 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 9.51iT - 53T^{2} \) |
| 59 | \( 1 + 0.428iT - 59T^{2} \) |
| 61 | \( 1 - 1.11iT - 61T^{2} \) |
| 67 | \( 1 + 2.35iT - 67T^{2} \) |
| 71 | \( 1 + 6.12T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 12.4iT - 89T^{2} \) |
| 97 | \( 1 + 8.04iT - 97T^{2} \) |
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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
−8.773364180696285621133394697380, −8.115834821267755604969787708616, −7.42135804054283443905140280782, −6.42176428773582321693157230897, −5.79202203416193630861216813950, −5.09595048067096958507684757074, −4.25607708417770555362138311032, −3.27208451284731473506940325666, −1.70459494217648759246507309254, −0.917321976110004157474147405869,
0.996535011532860407003201481051, 1.87944964263194770335189572183, 3.42996918038408259260312384111, 4.48895587238277888286601269762, 4.95624878421110219831102232160, 5.82487347808285846792108283871, 6.66526507325241735761584458164, 7.31578280131185085379244115198, 8.335320854111865173699801849020, 8.842253599988329848141220761247