| L(s) = 1 | + 2.53i·3-s − 31.0i·7-s + 20.5·9-s + 11·11-s − 79.1i·13-s − 54.4i·17-s − 105.·19-s + 78.8·21-s − 72.0i·23-s + 120. i·27-s − 228.·29-s + 22.7·31-s + 27.9i·33-s − 140. i·37-s + 201.·39-s + ⋯ |
| L(s) = 1 | + 0.488i·3-s − 1.67i·7-s + 0.761·9-s + 0.301·11-s − 1.68i·13-s − 0.776i·17-s − 1.27·19-s + 0.819·21-s − 0.653i·23-s + 0.860i·27-s − 1.46·29-s + 0.132·31-s + 0.147i·33-s − 0.624i·37-s + 0.825·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.123775867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123775867\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 2.53iT - 27T^{2} \) |
| 7 | \( 1 + 31.0iT - 343T^{2} \) |
| 13 | \( 1 + 79.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 54.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 228.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 22.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 140. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 62.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 481. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 69.1iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 453. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 225.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 320.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 200. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 971.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 688. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 544.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 788. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 842. iT - 9.12e5T^{2} \) |
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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
−9.367317718825131111281809382743, −8.153275531687693652417557338989, −7.45820211613666968506446685197, −6.77688844808574006176416843998, −5.63725468230956603068156520032, −4.47783794696925704564264749556, −4.02093066872797519393076560317, −2.92475755677558875260601969504, −1.28870530319547676419461106603, −0.26932951511772553823108687709,
1.78695922576799862318158775069, 2.07285914302726225534194578209, 3.69116141940300616569805157187, 4.61964148712638756077949145640, 5.78299584389586882416766145959, 6.45094978440253217498998710487, 7.21439202126780774098295157289, 8.336382963774411364187506061180, 8.989705869271898138073016463000, 9.578165058157278552342524502100
