L(s) = 1 | + (0.208 − 1.71i)3-s − 5-s + 0.187i·7-s + (−2.91 − 0.716i)9-s + 4.36·11-s + 6.75·13-s + (−0.208 + 1.71i)15-s − 0.956·17-s + 5.93i·19-s + (0.322 + 0.0390i)21-s + (4.02 − 2.60i)23-s + 25-s + (−1.83 + 4.85i)27-s − 4.44i·29-s − 3.97·31-s + ⋯ |
L(s) = 1 | + (0.120 − 0.992i)3-s − 0.447·5-s + 0.0708i·7-s + (−0.971 − 0.238i)9-s + 1.31·11-s + 1.87·13-s + (−0.0538 + 0.443i)15-s − 0.231·17-s + 1.36i·19-s + (0.0703 + 0.00852i)21-s + (0.839 − 0.542i)23-s + 0.200·25-s + (−0.353 + 0.935i)27-s − 0.825i·29-s − 0.714·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.793987407\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.793987407\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.208 + 1.71i)T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-4.02 + 2.60i)T \) |
good | 7 | \( 1 - 0.187iT - 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 + 0.956T + 17T^{2} \) |
| 19 | \( 1 - 5.93iT - 19T^{2} \) |
| 29 | \( 1 + 4.44iT - 29T^{2} \) |
| 31 | \( 1 + 3.97T + 31T^{2} \) |
| 37 | \( 1 - 9.51iT - 37T^{2} \) |
| 41 | \( 1 + 8.47iT - 41T^{2} \) |
| 43 | \( 1 + 7.63iT - 43T^{2} \) |
| 47 | \( 1 + 7.92iT - 47T^{2} \) |
| 53 | \( 1 - 6.24T + 53T^{2} \) |
| 59 | \( 1 - 0.525iT - 59T^{2} \) |
| 61 | \( 1 + 2.02iT - 61T^{2} \) |
| 67 | \( 1 + 7.87iT - 67T^{2} \) |
| 71 | \( 1 - 2.38iT - 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 + 13.4iT - 79T^{2} \) |
| 83 | \( 1 - 4.08T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−8.980600524987397587946936398265, −8.693041059111897807300719885100, −7.87077927743011231065610674930, −6.88627221866378549263359338003, −6.32556316816697862465993761411, −5.53581384628983151979471973022, −3.98382609756473828072479668656, −3.44412855357633381714337068276, −1.92175281528630338626049885850, −0.935513714152476071302327916409,
1.16711068157779886149143845328, 2.94599103915216169221146162287, 3.80878266317208377213252974588, 4.39196271000826870899345662127, 5.50552430725582737835401209564, 6.38915952280230522850644880314, 7.25892979011162994875019266869, 8.451811763120717701207030350843, 9.018116434217975278019657718781, 9.438952601715242957277832845676