L(s) = 1 | + 1.06·2-s − 3-s − 0.859·4-s + 0.128·5-s − 1.06·6-s − 7-s − 3.05·8-s + 9-s + 0.137·10-s − 1.51·11-s + 0.859·12-s + 7.04·13-s − 1.06·14-s − 0.128·15-s − 1.54·16-s + 1.04·17-s + 1.06·18-s − 7.25·19-s − 0.110·20-s + 21-s − 1.61·22-s − 9.19·23-s + 3.05·24-s − 4.98·25-s + 7.52·26-s − 27-s + 0.859·28-s + ⋯ |
L(s) = 1 | + 0.755·2-s − 0.577·3-s − 0.429·4-s + 0.0574·5-s − 0.435·6-s − 0.377·7-s − 1.07·8-s + 0.333·9-s + 0.0433·10-s − 0.457·11-s + 0.248·12-s + 1.95·13-s − 0.285·14-s − 0.0331·15-s − 0.385·16-s + 0.253·17-s + 0.251·18-s − 1.66·19-s − 0.0246·20-s + 0.218·21-s − 0.345·22-s − 1.91·23-s + 0.623·24-s − 0.996·25-s + 1.47·26-s − 0.192·27-s + 0.162·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.431117586\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431117586\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 5 | \( 1 - 0.128T + 5T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 - 7.04T + 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 + 9.19T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 5.99T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 6.56T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.01T + 53T^{2} \) |
| 59 | \( 1 - 1.21T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 - 0.339T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 - 0.441T + 83T^{2} \) |
| 89 | \( 1 - 8.73T + 89T^{2} \) |
| 97 | \( 1 - 6.25T + 97T^{2} \) |
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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
−8.396147177283530261194935186124, −7.86624022087262649554161943709, −6.38968089269680914796251554575, −6.17669719642592533441607377363, −5.65728186228150636631072526610, −4.44467379352582582172409224533, −4.11415128489437194552054396962, −3.25080147653720797169645248294, −2.06648284279058640597800171170, −0.62051319682527000971772657487,
0.62051319682527000971772657487, 2.06648284279058640597800171170, 3.25080147653720797169645248294, 4.11415128489437194552054396962, 4.44467379352582582172409224533, 5.65728186228150636631072526610, 6.17669719642592533441607377363, 6.38968089269680914796251554575, 7.86624022087262649554161943709, 8.396147177283530261194935186124