L(s) = 1 | + 3.40·3-s + 2.23i·5-s − 2.64i·7-s + 8.62·9-s + 5.55·11-s + 1.06i·13-s + 7.62i·15-s − 5.90·17-s − 9.02i·21-s − 5.00·25-s + 19.1·27-s + 4.83i·29-s + 18.9·33-s + 5.91·35-s + 3.62i·39-s + ⋯ |
L(s) = 1 | + 1.96·3-s + 0.999i·5-s − 0.999i·7-s + 2.87·9-s + 1.67·11-s + 0.294i·13-s + 1.96i·15-s − 1.43·17-s − 1.96i·21-s − 1.00·25-s + 3.68·27-s + 0.898i·29-s + 3.29·33-s + 0.999·35-s + 0.580i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.063370694\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.063370694\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 - 3.40T + 3T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 13 | \( 1 - 1.06iT - 13T^{2} \) |
| 17 | \( 1 + 5.90T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4.83iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 13.6iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 14.8iT - 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 3.45T + 97T^{2} \) |
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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
−9.001516143117055305624462164212, −8.436042345165817757270152402746, −7.43381484568710160227253766398, −6.90189569465115123556484661948, −6.52585484840970022084812603843, −4.54789612180504704643208246238, −3.86850954716957725545021171938, −3.40703344662885728384523092621, −2.31096406055067731660133060394, −1.47167848168330267057561296593,
1.37479427797207460040005254666, 2.15822320697179208719306949709, 3.09372476974719442973342705839, 4.18061945425152620928671950619, 4.53038559157730063426599977929, 5.96414024656214411177165856734, 6.82463321366154153253291967280, 7.78788443210456771932264941898, 8.509788782484018605942821377339, 8.968944268045352712821961586757