| L(s) = 1 | + 1.69i·5-s + (2.13 + 1.56i)7-s + 0.794i·11-s + 1.87i·13-s − 4.34i·17-s + 2.39·19-s − 3.62i·23-s + 2.12·25-s + 4.41·29-s + 1.87·31-s + (−2.64 + 3.62i)35-s + 3.12·37-s − 4.34i·41-s − 0.876i·43-s + 12.0·47-s + ⋯ |
| L(s) = 1 | + 0.758i·5-s + (0.807 + 0.590i)7-s + 0.239i·11-s + 0.519i·13-s − 1.05i·17-s + 0.550·19-s − 0.755i·23-s + 0.424·25-s + 0.820·29-s + 0.336·31-s + (−0.447 + 0.612i)35-s + 0.513·37-s − 0.678i·41-s − 0.133i·43-s + 1.76·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.251546258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.251546258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.13 - 1.56i)T \) |
| good | 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 11 | \( 1 - 0.794iT - 11T^{2} \) |
| 13 | \( 1 - 1.87iT - 13T^{2} \) |
| 17 | \( 1 + 4.34iT - 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 + 3.62iT - 23T^{2} \) |
| 29 | \( 1 - 4.41T + 29T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 4.34iT - 41T^{2} \) |
| 43 | \( 1 + 0.876iT - 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 0.876iT - 67T^{2} \) |
| 71 | \( 1 + 5.21iT - 71T^{2} \) |
| 73 | \( 1 - 3.74iT - 73T^{2} \) |
| 79 | \( 1 + 3.12iT - 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 7.73iT - 89T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
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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
−8.706213643936132703122086137959, −7.66277506978620836010757614376, −7.16778957547256549753988294285, −6.41954139084685697928514095263, −5.55237868692158646819570658547, −4.80812373384978249199282257016, −4.06440732793581255919227632797, −2.79817150428810234356996799116, −2.38202465550598188687703857530, −1.03879054892189823637846828069,
0.819231195782499946019465881789, 1.55403650829920865498894098673, 2.83531753233004597365731338185, 3.87361384150531990667551478280, 4.55657212934930484505961591165, 5.31408442551790763910519788418, 5.97290835971870951085634495411, 6.98305856755494905292266156795, 7.75824867716090431919691118789, 8.306242951893085978184408455968
