L(s) = 1 | + (2.25 − 2.25i)3-s + (1.48 − 1.48i)7-s − 7.15i·9-s + 0.806·11-s + (−0.437 + 0.437i)13-s + (−3.15 + 3.15i)17-s + (−1.81 − 3.96i)19-s − 6.67i·21-s + (−1.86 − 1.86i)23-s + (−9.36 − 9.36i)27-s + 4.50·29-s − 6.67i·31-s + (1.81 − 1.81i)33-s + (5.29 + 5.29i)37-s + 1.96i·39-s + ⋯ |
L(s) = 1 | + (1.30 − 1.30i)3-s + (0.559 − 0.559i)7-s − 2.38i·9-s + 0.243·11-s + (−0.121 + 0.121i)13-s + (−0.765 + 0.765i)17-s + (−0.416 − 0.909i)19-s − 1.45i·21-s + (−0.389 − 0.389i)23-s + (−1.80 − 1.80i)27-s + 0.836·29-s − 1.19i·31-s + (0.316 − 0.316i)33-s + (0.870 + 0.870i)37-s + 0.315i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.746718618\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.746718618\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (1.81 + 3.96i)T \) |
good | 3 | \( 1 + (-2.25 + 2.25i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.48 + 1.48i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.806T + 11T^{2} \) |
| 13 | \( 1 + (0.437 - 0.437i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.15 - 3.15i)T - 17iT^{2} \) |
| 23 | \( 1 + (1.86 + 1.86i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 + 6.67iT - 31T^{2} \) |
| 37 | \( 1 + (-5.29 - 5.29i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.1iT - 41T^{2} \) |
| 43 | \( 1 + (-3.86 - 3.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.83 - 6.83i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.92 + 8.92i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 + (-4.94 - 4.94i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.04iT - 71T^{2} \) |
| 73 | \( 1 + (-6.19 - 6.19i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 + (5.32 + 5.32i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + (-7.46 - 7.46i)T + 97iT^{2} \) |
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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
−8.764657470639789883287211833035, −8.077551570603514105172152781516, −7.55604897576137543973656363264, −6.68608166673907823820427916318, −6.19383951005395139284095093118, −4.59570532983983841745120693290, −3.84703608253829208115208220293, −2.66042632193977265496396930178, −1.96105696225305635015702470470, −0.841961934614638315574179660472,
1.88769515930737701205905740074, 2.75674818325843326679430523100, 3.63446730071075183097917209916, 4.53939084511276670732124394311, 5.08297863434891827136342865309, 6.20922933560992424080160474851, 7.47685609964719369025545615001, 8.183131385499609371944176695426, 8.796165804222218383795697075403, 9.346662691463728347564552569005