| L(s) = 1 | + 0.939·3-s − 1.90i·7-s − 2.11·9-s − 2.70i·11-s + 5.75·13-s − 6.65i·17-s + 7.45i·19-s − 1.78i·21-s − 0.220i·23-s − 4.80·27-s − 4.62i·29-s + 5.76·31-s − 2.54i·33-s + 3.84·37-s + 5.40·39-s + ⋯ |
| L(s) = 1 | + 0.542·3-s − 0.718i·7-s − 0.705·9-s − 0.816i·11-s + 1.59·13-s − 1.61i·17-s + 1.71i·19-s − 0.389i·21-s − 0.0460i·23-s − 0.925·27-s − 0.858i·29-s + 1.03·31-s − 0.442i·33-s + 0.631·37-s + 0.866·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.893780925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.893780925\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.939T + 3T^{2} \) |
| 7 | \( 1 + 1.90iT - 7T^{2} \) |
| 11 | \( 1 + 2.70iT - 11T^{2} \) |
| 13 | \( 1 - 5.75T + 13T^{2} \) |
| 17 | \( 1 + 6.65iT - 17T^{2} \) |
| 19 | \( 1 - 7.45iT - 19T^{2} \) |
| 23 | \( 1 + 0.220iT - 23T^{2} \) |
| 29 | \( 1 + 4.62iT - 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 - 3.84T + 37T^{2} \) |
| 41 | \( 1 + 7.94T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 3.09T + 53T^{2} \) |
| 59 | \( 1 + 5.84iT - 59T^{2} \) |
| 61 | \( 1 + 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 5.42T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 0.290iT - 73T^{2} \) |
| 79 | \( 1 - 0.354T + 79T^{2} \) |
| 83 | \( 1 + 6.09T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 + 4.56iT - 97T^{2} \) |
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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
−8.289695222796190468873951757314, −7.77117606757400693635253744991, −6.72644042314634762557727161118, −6.05788709597503336598916312873, −5.37659353113220329055261298903, −4.24737357852047744413683973699, −3.46984831601876945892241829872, −2.95184813023533195963059492292, −1.63769700447174123732600010927, −0.51232105410223067258114410832,
1.34368856260675667863694771799, 2.33698606593799472661192664619, 3.14327026904573963848826041789, 3.96111520846372738742720166447, 4.90126262050381607639024534017, 5.78347866603507020274545538478, 6.39204176793346681640410525562, 7.17320189998129566408721519255, 8.251592291331737842387369124574, 8.670419966673517039775976522712
