| L(s) = 1 | + 3.07i·3-s + 1.34i·5-s − 0.0167i·7-s − 6.44·9-s − 2.90i·11-s + 0.0774·13-s − 4.12·15-s + (2.86 − 2.96i)17-s + 7.12·19-s + 0.0514·21-s − 2.37i·23-s + 3.20·25-s − 10.5i·27-s − 3.11i·29-s − 7.22i·31-s + ⋯ |
| L(s) = 1 | + 1.77i·3-s + 0.599i·5-s − 0.00632i·7-s − 2.14·9-s − 0.874i·11-s + 0.0214·13-s − 1.06·15-s + (0.694 − 0.719i)17-s + 1.63·19-s + 0.0112·21-s − 0.494i·23-s + 0.640·25-s − 2.03i·27-s − 0.579i·29-s − 1.29i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.806118167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.806118167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-2.86 + 2.96i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 3.07iT - 3T^{2} \) |
| 5 | \( 1 - 1.34iT - 5T^{2} \) |
| 7 | \( 1 + 0.0167iT - 7T^{2} \) |
| 11 | \( 1 + 2.90iT - 11T^{2} \) |
| 13 | \( 1 - 0.0774T + 13T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 2.37iT - 23T^{2} \) |
| 29 | \( 1 + 3.11iT - 29T^{2} \) |
| 31 | \( 1 + 7.22iT - 31T^{2} \) |
| 37 | \( 1 + 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 9.14iT - 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + 5.09T + 53T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 + 6.23iT - 71T^{2} \) |
| 73 | \( 1 + 0.615iT - 73T^{2} \) |
| 79 | \( 1 + 3.90iT - 79T^{2} \) |
| 83 | \( 1 - 2.13T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 10.5iT - 97T^{2} \) |
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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
−8.839543116718093809388441819468, −7.86442699687198370208599949128, −7.18403862904425886334367609179, −5.91269402443001951576435027195, −5.60030721641976699986935547163, −4.70539147998412899878311709568, −3.90558691880061264280047489765, −3.19066442416285288948415841836, −2.62956353263587670722304850420, −0.61223415583842025359601025093,
1.13229373331158610387322866197, 1.44030327545229088040631776546, 2.65008313405461984802501734882, 3.47980149174350185707651023565, 4.88227360502130626400877185243, 5.42627097345344652631391150367, 6.34916909956124121236721886458, 6.94272756608331098062366084620, 7.72128813716212969582164527738, 7.999560177122960351381801795675
