| L(s) = 1 | − 5-s + 4.64i·7-s + 3.84·11-s − 6.83·13-s − 4.29·17-s − 6.20i·19-s + (2.23 + 4.24i)23-s + 25-s − 1.90i·29-s + 5.07·31-s − 4.64i·35-s + 2.26i·37-s + 10.8i·41-s − 3.10i·43-s − 3.10i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.75i·7-s + 1.16·11-s − 1.89·13-s − 1.04·17-s − 1.42i·19-s + (0.465 + 0.885i)23-s + 0.200·25-s − 0.353i·29-s + 0.910·31-s − 0.785i·35-s + 0.372i·37-s + 1.69i·41-s − 0.474i·43-s − 0.452i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3852139896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3852139896\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-2.23 - 4.24i)T \) |
| good | 7 | \( 1 - 4.64iT - 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 6.83T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 + 6.20iT - 19T^{2} \) |
| 29 | \( 1 + 1.90iT - 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 - 2.26iT - 37T^{2} \) |
| 41 | \( 1 - 10.8iT - 41T^{2} \) |
| 43 | \( 1 + 3.10iT - 43T^{2} \) |
| 47 | \( 1 + 3.10iT - 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 + 8.43iT - 59T^{2} \) |
| 61 | \( 1 + 8.26iT - 61T^{2} \) |
| 67 | \( 1 - 12.6iT - 67T^{2} \) |
| 71 | \( 1 - 10.5iT - 71T^{2} \) |
| 73 | \( 1 + 5.77T + 73T^{2} \) |
| 79 | \( 1 - 2.14iT - 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 2.95T + 89T^{2} \) |
| 97 | \( 1 + 3.87iT - 97T^{2} \) |
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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
−7.59395427254856676063155281827, −6.80187900202195149184565553426, −6.42811813225367282067921024723, −5.39317622166987179363941993979, −4.84907172913961990765620448678, −4.26084102806689099513469619916, −2.94433449934479477334050862917, −2.61536906276129318127018955288, −1.63704729800727634906097947501, −0.10394445572308699796525927407,
0.898233817309746459372985431230, 1.91439539488136164435456088865, 3.01954309369732840306246593845, 3.91154018536280956245595686378, 4.39533113055396350597134806364, 4.89105879649014074602271287524, 6.15082453198713537111788128946, 6.76887345531961576752202824548, 7.35382852216377700200683101249, 7.71388269628924626830665634491
