L(s) = 1 | + 2·4-s + 3.60i·5-s − 3·9-s + 3.60i·13-s + 4·16-s + 3.60i·19-s + 7.21i·20-s + 23-s − 7.99·25-s − 5·29-s − 10.8i·31-s − 6·36-s + 7.21i·41-s + 9·43-s − 10.8i·45-s + ⋯ |
L(s) = 1 | + 4-s + 1.61i·5-s − 9-s + 0.999i·13-s + 16-s + 0.827i·19-s + 1.61i·20-s + 0.208·23-s − 1.59·25-s − 0.928·29-s − 1.94i·31-s − 36-s + 1.12i·41-s + 1.37·43-s − 1.61i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15548 + 1.15548i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15548 + 1.15548i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - 3.60iT \) |
good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 3.60iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.60iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 10.8iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 7.21iT - 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 - 3.60iT - 47T^{2} \) |
| 53 | \( 1 - 11T + 53T^{2} \) |
| 59 | \( 1 - 14.4iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 + 18.0iT - 83T^{2} \) |
| 89 | \( 1 - 3.60iT - 89T^{2} \) |
| 97 | \( 1 + 18.0iT - 97T^{2} \) |
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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
−10.95501419740165719230398927607, −10.17301053760386419033308267863, −9.177948290762569783891091012030, −7.82472286374025567874595526250, −7.26779433921307144355118806726, −6.24538692255915754878503141910, −5.85277215831893289814021448477, −3.93798235111331440903582746562, −2.90183652915865826561172861350, −2.11003093269110094512873523839,
0.865643589004246538352849698417, 2.36122142279252356623599827622, 3.60556027641073429976861898300, 5.18648095530453650072178156842, 5.54240797548505922390033308921, 6.80306429687961310874924384934, 7.87888024039073588277570836718, 8.599949348034413221895450657824, 9.310666724135831601117130001293, 10.53758706961148283263872825775