L(s) = 1 | + 2.76·5-s + (−0.480 + 2.60i)7-s − 5.75·11-s + 2·13-s − 6.71i·17-s − 5.20i·19-s − 4.43i·23-s + 2.66·25-s + 1.54i·29-s − 8.05·31-s + (−1.33 + 7.20i)35-s + 4.42i·37-s + 0.209i·41-s − 10.5·43-s + 4.58·47-s + ⋯ |
L(s) = 1 | + 1.23·5-s + (−0.181 + 0.983i)7-s − 1.73·11-s + 0.554·13-s − 1.62i·17-s − 1.19i·19-s − 0.924i·23-s + 0.533·25-s + 0.286i·29-s − 1.44·31-s + (−0.225 + 1.21i)35-s + 0.727i·37-s + 0.0326i·41-s − 1.60·43-s + 0.669·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.033441209\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033441209\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.480 - 2.60i)T \) |
good | 5 | \( 1 - 2.76T + 5T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6.71iT - 17T^{2} \) |
| 19 | \( 1 + 5.20iT - 19T^{2} \) |
| 23 | \( 1 + 4.43iT - 23T^{2} \) |
| 29 | \( 1 - 1.54iT - 29T^{2} \) |
| 31 | \( 1 + 8.05T + 31T^{2} \) |
| 37 | \( 1 - 4.42iT - 37T^{2} \) |
| 41 | \( 1 - 0.209iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 + 8.05iT - 53T^{2} \) |
| 59 | \( 1 + 5.53iT - 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 4.04T + 67T^{2} \) |
| 71 | \( 1 - 1.10iT - 71T^{2} \) |
| 73 | \( 1 - 9.59iT - 73T^{2} \) |
| 79 | \( 1 + 14.7iT - 79T^{2} \) |
| 83 | \( 1 - 8.86iT - 83T^{2} \) |
| 89 | \( 1 + 13.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.58iT - 97T^{2} \) |
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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
−8.396189662616573408137460068561, −7.41041024840888055307816951218, −6.65767312897578531330537454154, −5.89714548733922432877902197353, −5.12042339362848870934334143614, −4.92901647372317068830332305813, −3.17278347497016499469456710426, −2.60517567494007918950111362801, −1.90711247947262028754099243888, −0.26332106281570304303525063134,
1.42889940851340566984453041599, 2.09765569965617703909680316887, 3.32551604271545992745387387674, 3.98695242246424937911409404379, 5.13881480782538036290456628320, 5.82611836953688349254290369340, 6.20806995128437093257385318823, 7.37477127938161980279377202913, 7.84361777366613028594583737142, 8.629376904385313910025515893550