| L(s) = 1 | + 0.618i·3-s + 1.61i·7-s + 2.61·9-s + 5.52·11-s + 5.52i·13-s + 3.41i·17-s + 3.41·19-s − 1.00·21-s + 2.38i·23-s + 3.47i·27-s − 1.09·29-s − 5.52·31-s + 3.41i·33-s − 8.93i·37-s − 3.41·39-s + ⋯ |
| L(s) = 1 | + 0.356i·3-s + 0.611i·7-s + 0.872·9-s + 1.66·11-s + 1.53i·13-s + 0.827i·17-s + 0.782·19-s − 0.218·21-s + 0.496i·23-s + 0.668i·27-s − 0.202·29-s − 0.991·31-s + 0.593i·33-s − 1.46i·37-s − 0.546·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.328368430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328368430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.618iT - 3T^{2} \) |
| 7 | \( 1 - 1.61iT - 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 - 5.52iT - 13T^{2} \) |
| 17 | \( 1 - 3.41iT - 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 2.38iT - 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + 5.52T + 31T^{2} \) |
| 37 | \( 1 + 8.93iT - 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 - 12.3iT - 43T^{2} \) |
| 47 | \( 1 - 1.09iT - 47T^{2} \) |
| 53 | \( 1 + 11.0iT - 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + 12.3iT - 73T^{2} \) |
| 79 | \( 1 - 8.93T + 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−8.963393299452781935667377856807, −7.85136071914670938040824454758, −7.08777880405693755402918917027, −6.44348902202337300914775722607, −5.76303913242643614664607163824, −4.69511331903703308376712104828, −4.04015080520823707400154702068, −3.47280120324780030742893062310, −1.97765129691247100050843085234, −1.38215660771820369211985995397,
0.75630258947216390081934954928, 1.42773235487355223101937324935, 2.77037119646334292503654871419, 3.70559380428762603918061498334, 4.34758807551704050676974139014, 5.32247078901201741780972599418, 6.12943565287008846522703150540, 7.07235657287987615480421537833, 7.29430018264624810769715635263, 8.148709059447746444761229278644
