L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s + 1.75i·7-s − i·8-s + 9-s − 10-s + 1.19i·11-s − 12-s − 1.75·14-s + i·15-s + 16-s + 6.76·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 0.662i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.361i·11-s − 0.288·12-s − 0.468·14-s + 0.258i·15-s + 0.250·16-s + 1.64·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.531758840\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.531758840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.75iT - 7T^{2} \) |
| 11 | \( 1 - 1.19iT - 11T^{2} \) |
| 17 | \( 1 - 6.76T + 17T^{2} \) |
| 19 | \( 1 + 8.63iT - 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 + 6.80iT - 31T^{2} \) |
| 37 | \( 1 + 4.34iT - 37T^{2} \) |
| 41 | \( 1 + 6.63iT - 41T^{2} \) |
| 43 | \( 1 + 4.07T + 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 13.7iT - 59T^{2} \) |
| 61 | \( 1 + 1.18T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 - 2.32iT - 71T^{2} \) |
| 73 | \( 1 - 1.59iT - 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 + 8.40iT - 83T^{2} \) |
| 89 | \( 1 - 4.49iT - 89T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−8.308062445000546880053781041905, −7.40525833281566744486141942373, −7.12785222183625608529665987711, −6.25285386716315589201985654803, −5.39166804810466838771265038870, −4.84168437194574718962993812089, −3.79966808323726158849833781461, −2.96564075642220655788875884893, −2.25270437277617276257157482121, −0.77554403893384420728337870847,
1.07481664061439415025040204479, 1.51037591269071385119672052316, 3.01903863790114294665378567774, 3.37317424653556137921514007711, 4.26458949944005174961018015484, 5.04936534958229252930142728427, 5.82409169619316438657609604269, 6.78904837768730958776016128326, 7.75551774705680030929924503061, 8.162139766708328005131381227726