L(s) = 1 | + 3-s + (3.08 − 3.08i)5-s + (−2.85 − 2.85i)7-s + 9-s + (−2.57 + 2.57i)11-s + (−3.51 − 0.821i)13-s + (3.08 − 3.08i)15-s − 1.07i·17-s + (−1.46 − 1.46i)19-s + (−2.85 − 2.85i)21-s + 5.52·23-s − 14.0i·25-s + 27-s − 0.512i·29-s + (2.85 − 2.85i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (1.38 − 1.38i)5-s + (−1.08 − 1.08i)7-s + 0.333·9-s + (−0.776 + 0.776i)11-s + (−0.973 − 0.227i)13-s + (0.797 − 0.797i)15-s − 0.259i·17-s + (−0.335 − 0.335i)19-s + (−0.623 − 0.623i)21-s + 1.15·23-s − 2.81i·25-s + 0.192·27-s − 0.0951i·29-s + (0.513 − 0.513i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.869080720\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.869080720\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (3.51 + 0.821i)T \) |
good | 5 | \( 1 + (-3.08 + 3.08i)T - 5iT^{2} \) |
| 7 | \( 1 + (2.85 + 2.85i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.57 - 2.57i)T - 11iT^{2} \) |
| 17 | \( 1 + 1.07iT - 17T^{2} \) |
| 19 | \( 1 + (1.46 + 1.46i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 + 0.512iT - 29T^{2} \) |
| 31 | \( 1 + (-2.85 + 2.85i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.97 - 1.97i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.0336 + 0.0336i)T + 41iT^{2} \) |
| 43 | \( 1 - 3.49iT - 43T^{2} \) |
| 47 | \( 1 + (6.45 + 6.45i)T + 47iT^{2} \) |
| 53 | \( 1 - 8.04iT - 53T^{2} \) |
| 59 | \( 1 + (1.23 - 1.23i)T - 59iT^{2} \) |
| 61 | \( 1 + 4.66iT - 61T^{2} \) |
| 67 | \( 1 + (2.03 + 2.03i)T + 67iT^{2} \) |
| 71 | \( 1 + (-5.60 + 5.60i)T - 71iT^{2} \) |
| 73 | \( 1 + (5.73 - 5.73i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.65iT - 79T^{2} \) |
| 83 | \( 1 + (0.153 + 0.153i)T + 83iT^{2} \) |
| 89 | \( 1 + (-11.3 + 11.3i)T - 89iT^{2} \) |
| 97 | \( 1 + (-8.80 - 8.80i)T + 97iT^{2} \) |
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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
−9.590201401567458392902182194597, −8.858159135429468703668916619837, −7.82118585221197838834155744949, −7.01484635288665136791068016227, −6.12222706384929198581782002405, −4.95430655818301719301902583256, −4.53203721730973590854047192024, −3.02025480384066692997339680931, −2.06293831512490755930669996022, −0.68308781944206107030960918597,
2.08515917046041978592066661943, 2.80520295222023060561906627061, 3.28668951449670254489065645337, 5.13603312135259125731655564621, 5.94361670198691578636137303303, 6.56586606098372463866600159729, 7.32805059918313133883750811510, 8.526153125759623457974539287314, 9.326226807561658996348344622949, 9.874422085701125620216844714619