| L(s) = 1 | − i·2-s − 4-s + 0.260·5-s + (−1.84 − 1.89i)7-s + i·8-s − 0.260i·10-s − i·11-s + 5.36i·13-s + (−1.89 + 1.84i)14-s + 16-s − 4.05·17-s − 0.629i·19-s − 0.260·20-s − 22-s − 4.93·25-s + 5.36·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.116·5-s + (−0.697 − 0.716i)7-s + 0.353i·8-s − 0.0824i·10-s − 0.301i·11-s + 1.48i·13-s + (−0.506 + 0.493i)14-s + 0.250·16-s − 0.983·17-s − 0.144i·19-s − 0.0583·20-s − 0.213·22-s − 0.986·25-s + 1.05·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5086119401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5086119401\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (1.84 + 1.89i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.260T + 5T^{2} \) |
| 13 | \( 1 - 5.36iT - 13T^{2} \) |
| 17 | \( 1 + 4.05T + 17T^{2} \) |
| 19 | \( 1 + 0.629iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 4.94iT - 29T^{2} \) |
| 31 | \( 1 - 4.42iT - 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 - 9.78T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 + 5.30iT - 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 5.62T + 79T^{2} \) |
| 83 | \( 1 - 0.0507T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 0.737iT - 97T^{2} \) |
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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
−9.811773408955713430953103952111, −9.081259935499761712133911781749, −8.442910827898572040165123186780, −7.10340955588512608325077149471, −6.66886028916196487425860489054, −5.50765715525273552343522168877, −4.35643713156524773116495602798, −3.77090483858523314030523462340, −2.60820808513019615143494836712, −1.44433532799006207901965941232,
0.20463531435619907560049729529, 2.21200436451613776363622875608, 3.31333759233919032801179293922, 4.40432909418737817151922984312, 5.51414193291617682883091602335, 6.01088978861878671887220558284, 6.88166678170281359658622660354, 7.84372761421734499074050523276, 8.452865955824035029398003732762, 9.419608186780822777731470420929
