L(s) = 1 | − 3.91i·5-s + (−2.03 − 1.68i)7-s + 1.62·11-s + 2.54·13-s − 2.94·17-s − 2.72·19-s + 8.57i·23-s − 10.3·25-s − 8.21·29-s − 8.08i·31-s + (−6.61 + 7.98i)35-s − 8.05i·37-s − 9.17·41-s + 9.05i·43-s + 1.17·47-s + ⋯ |
L(s) = 1 | − 1.75i·5-s + (−0.769 − 0.638i)7-s + 0.490·11-s + 0.706·13-s − 0.713·17-s − 0.624·19-s + 1.78i·23-s − 2.07·25-s − 1.52·29-s − 1.45i·31-s + (−1.11 + 1.34i)35-s − 1.32i·37-s − 1.43·41-s + 1.38i·43-s + 0.171·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2341052464\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2341052464\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.03 + 1.68i)T \) |
good | 5 | \( 1 + 3.91iT - 5T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 - 8.57iT - 23T^{2} \) |
| 29 | \( 1 + 8.21T + 29T^{2} \) |
| 31 | \( 1 + 8.08iT - 31T^{2} \) |
| 37 | \( 1 + 8.05iT - 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 - 9.05iT - 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 + 1.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.74T + 61T^{2} \) |
| 67 | \( 1 - 7.35iT - 67T^{2} \) |
| 71 | \( 1 + 7.71iT - 71T^{2} \) |
| 73 | \( 1 + 15.0iT - 73T^{2} \) |
| 79 | \( 1 - 0.0913T + 79T^{2} \) |
| 83 | \( 1 - 2.03iT - 83T^{2} \) |
| 89 | \( 1 + 1.48T + 89T^{2} \) |
| 97 | \( 1 - 7.53iT - 97T^{2} \) |
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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
−7.985143218192542195434883896631, −7.36098338260433734284716209213, −6.33822609480120802850132183295, −5.73247533926567980173429281194, −4.92735862584144447447175895480, −3.92141205299820860374067657725, −3.74126200313296960830957399555, −2.03544259369734398713381177450, −1.15164098924451936868759952849, −0.06864700108348249314180359131,
1.91726810573994952928823689077, 2.71309296219594986967570651296, 3.43048690626935984033008492814, 4.13662458564826711766543328430, 5.41930311096074300273705974153, 6.25750217892847609421707704143, 6.78391114067669533653429876446, 7.02574432778858691650714846244, 8.456506399987363716263841846385, 8.707524927862060868018892308948