L(s) = 1 | + (0.340 + 2.21i)5-s + (−3.49 − 3.49i)7-s + 1.20i·11-s + (0.0289 − 0.0289i)13-s + (4.88 − 4.88i)17-s + 6.86i·19-s + (0.707 + 0.707i)23-s + (−4.76 + 1.50i)25-s − 2.41·29-s + 4.34·31-s + (6.53 − 8.91i)35-s + (−7.27 − 7.27i)37-s − 1.65i·41-s + (−2.38 + 2.38i)43-s + (−1.54 + 1.54i)47-s + ⋯ |
L(s) = 1 | + (0.152 + 0.988i)5-s + (−1.32 − 1.32i)7-s + 0.363i·11-s + (0.00802 − 0.00802i)13-s + (1.18 − 1.18i)17-s + 1.57i·19-s + (0.147 + 0.147i)23-s + (−0.953 + 0.300i)25-s − 0.449·29-s + 0.780·31-s + (1.10 − 1.50i)35-s + (−1.19 − 1.19i)37-s − 0.258i·41-s + (−0.362 + 0.362i)43-s + (−0.226 + 0.226i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.231674743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.231674743\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.340 - 2.21i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (3.49 + 3.49i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.20iT - 11T^{2} \) |
| 13 | \( 1 + (-0.0289 + 0.0289i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.88 + 4.88i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.86iT - 19T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 + (7.27 + 7.27i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.65iT - 41T^{2} \) |
| 43 | \( 1 + (2.38 - 2.38i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.54 - 1.54i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.188 - 0.188i)T + 53iT^{2} \) |
| 59 | \( 1 - 0.182T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + (4.11 + 4.11i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.99iT - 71T^{2} \) |
| 73 | \( 1 + (-5.12 + 5.12i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.4iT - 79T^{2} \) |
| 83 | \( 1 + (7.95 + 7.95i)T + 83iT^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + (-3.45 - 3.45i)T + 97iT^{2} \) |
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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
−8.008997873641722066908036637826, −7.37510785144790714154099496027, −6.98046516232165042140170992397, −6.19814964640222858099031371579, −5.51642888747927919410799007151, −4.31512626874748654263038847411, −3.40439273188286953123312851491, −3.17518748218903022600031513914, −1.80048419705120287921673595704, −0.42554274131814503852644786690,
0.929509543507700545951198730053, 2.20465672493565387897974280272, 3.07930153404024590828079191940, 3.85650049156827672359256537123, 5.06977931497249170157695133839, 5.50505309481326654427280746101, 6.28640828145436933235573481201, 6.86433295805456651864454028822, 8.121075181954555347898393589403, 8.612412925138177567956933190627