L(s) = 1 | + 1.18·3-s − 0.0964·5-s − 1.78·7-s − 1.59·9-s + 1.62·11-s − 0.858·13-s − 0.114·15-s − 0.953·17-s + 3.91·19-s − 2.11·21-s − 1.65·23-s − 4.99·25-s − 5.44·27-s + 8.35·29-s − 6.03·31-s + 1.92·33-s + 0.171·35-s + 10.3·37-s − 1.01·39-s − 5.38·41-s − 1.55·43-s + 0.153·45-s + 0.729·47-s − 3.81·49-s − 1.12·51-s + 2.02·53-s − 0.156·55-s + ⋯ |
L(s) = 1 | + 0.683·3-s − 0.0431·5-s − 0.674·7-s − 0.532·9-s + 0.489·11-s − 0.238·13-s − 0.0294·15-s − 0.231·17-s + 0.898·19-s − 0.461·21-s − 0.344·23-s − 0.998·25-s − 1.04·27-s + 1.55·29-s − 1.08·31-s + 0.334·33-s + 0.0290·35-s + 1.70·37-s − 0.162·39-s − 0.840·41-s − 0.237·43-s + 0.0229·45-s + 0.106·47-s − 0.545·49-s − 0.158·51-s + 0.278·53-s − 0.0211·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.000705023\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.000705023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 + 0.0964T + 5T^{2} \) |
| 7 | \( 1 + 1.78T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 + 0.858T + 13T^{2} \) |
| 17 | \( 1 + 0.953T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 + 1.65T + 23T^{2} \) |
| 29 | \( 1 - 8.35T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 + 1.55T + 43T^{2} \) |
| 47 | \( 1 - 0.729T + 47T^{2} \) |
| 53 | \( 1 - 2.02T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 6.10T + 61T^{2} \) |
| 67 | \( 1 + 3.88T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 3.04T + 79T^{2} \) |
| 83 | \( 1 + 0.674T + 83T^{2} \) |
| 89 | \( 1 + 6.71T + 89T^{2} \) |
| 97 | \( 1 - 5.60T + 97T^{2} \) |
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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
−7.983528731546025726296700005214, −7.16154777175338343534941274854, −6.49377259639792312687106710912, −5.81172510429059115577539809871, −5.07132785120936502818853925478, −4.05283145743056227555781226062, −3.46515207647898826652309155139, −2.73670164635306109900267266046, −1.96697435916831042672006555201, −0.65578011700716576167069058225,
0.65578011700716576167069058225, 1.96697435916831042672006555201, 2.73670164635306109900267266046, 3.46515207647898826652309155139, 4.05283145743056227555781226062, 5.07132785120936502818853925478, 5.81172510429059115577539809871, 6.49377259639792312687106710912, 7.16154777175338343534941274854, 7.983528731546025726296700005214