L(s) = 1 | + (0.809 + 0.587i)2-s + (−1.02 + 3.16i)3-s + (0.309 + 0.951i)4-s + (2.47 − 1.79i)5-s + (−2.69 + 1.95i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−6.55 − 4.76i)9-s + 3.05·10-s + (−2.60 − 2.04i)11-s − 3.33·12-s + (1.62 + 1.18i)13-s + (−0.309 + 0.951i)14-s + (3.15 + 9.69i)15-s + (−0.809 + 0.587i)16-s + (3.13 − 2.27i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.594 + 1.82i)3-s + (0.154 + 0.475i)4-s + (1.10 − 0.804i)5-s + (−1.10 + 0.799i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (−2.18 − 1.58i)9-s + 0.967·10-s + (−0.786 − 0.617i)11-s − 0.962·12-s + (0.451 + 0.327i)13-s + (−0.0825 + 0.254i)14-s + (0.813 + 2.50i)15-s + (−0.202 + 0.146i)16-s + (0.759 − 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.830879 + 1.12717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.830879 + 1.12717i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (2.60 + 2.04i)T \) |
good | 3 | \( 1 + (1.02 - 3.16i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-2.47 + 1.79i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-1.62 - 1.18i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.13 + 2.27i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.0995 + 0.306i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 0.774T + 23T^{2} \) |
| 29 | \( 1 + (-0.0934 - 0.287i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.86 - 2.80i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.66 + 5.10i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.88 - 5.80i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8.28T + 43T^{2} \) |
| 47 | \( 1 + (-0.376 + 1.15i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.52 - 1.83i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (4.16 + 12.8i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (9.57 - 6.95i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 + (-11.0 + 7.99i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.70 - 5.26i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (14.2 + 10.3i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.249 + 0.181i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 4.69T + 89T^{2} \) |
| 97 | \( 1 + (8.33 + 6.05i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.47966747400538382685553367262, −12.20126771365872356855050752107, −11.19499407681121055260965198688, −10.17046233346085740534347091439, −9.292039865957692367531135224173, −8.441017081293933950643630555178, −6.15605438146593236722559830300, −5.38024899945685105466735002249, −4.75718926117945437472167195868, −3.20380010543030768510856674837,
1.59886887730278695533608751329, 2.77566173915414623639593032887, 5.33968262747354690819818429042, 6.18104292330237864838874865206, 7.03912183758740366310396806630, 8.111630895608175735822580981547, 10.11611650126991381219765793249, 10.81812422809816805387863645825, 11.91975487317799286166665127042, 12.79615909116695951001466586517