L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s + 1.69i·7-s − i·8-s + 9-s − 10-s − 2.15i·11-s + 12-s − 1.69·14-s − i·15-s + 16-s − 2.35·17-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 + 0.639i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.650i·11-s + 0.288·12-s − 0.452·14-s − 0.258i·15-s + 0.250·16-s − 0.571·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.007077148\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.007077148\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.69iT - 7T^{2} \) |
| 11 | \( 1 + 2.15iT - 11T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 - 0.198iT - 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 - 1.44iT - 31T^{2} \) |
| 37 | \( 1 + 0.801iT - 37T^{2} \) |
| 41 | \( 1 + 1.89iT - 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 8.87iT - 47T^{2} \) |
| 53 | \( 1 + 1.00T + 53T^{2} \) |
| 59 | \( 1 + 3.73iT - 59T^{2} \) |
| 61 | \( 1 + 6.32T + 61T^{2} \) |
| 67 | \( 1 + 7.56iT - 67T^{2} \) |
| 71 | \( 1 + 4.18iT - 71T^{2} \) |
| 73 | \( 1 + 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 - 8.43iT - 83T^{2} \) |
| 89 | \( 1 - 2.41iT - 89T^{2} \) |
| 97 | \( 1 + 0.0881iT - 97T^{2} \) |
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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
−8.205989492928119290155535959835, −7.38861783402956083639687909114, −6.59930065085825128862335316072, −6.25491304195658427451508683299, −5.33205840067449704884021827834, −4.90013502628408115834863376116, −3.79689068299856009326087889439, −2.99443411268074903542806114506, −1.82666995166732522605061441393, −0.38339072863749065379971307672,
0.854798684823446662482882527467, 1.71894822255597981223504137115, 2.80257865950877720234005857678, 3.83833589739774039399396316305, 4.57431409359409685413236435977, 5.01830314155687045643724091887, 5.99569165707473943228000165010, 6.82918955875806403263267639958, 7.46609189306214587646881017004, 8.328583871964921893755762157264