L(s) = 1 | + 1.06·3-s + 1.60·5-s − 0.812·7-s − 1.87·9-s + 5.39·11-s − 13-s + 1.69·15-s + 3.51·17-s − 3.73·19-s − 0.861·21-s − 7.75·23-s − 2.43·25-s − 5.16·27-s + 29-s + 6.56·31-s + 5.71·33-s − 1.30·35-s − 3.76·37-s − 1.06·39-s + 8.55·41-s + 12.8·43-s − 3.00·45-s + 11.4·47-s − 6.33·49-s + 3.72·51-s + 6.10·53-s + 8.63·55-s + ⋯ |
L(s) = 1 | + 0.612·3-s + 0.716·5-s − 0.307·7-s − 0.625·9-s + 1.62·11-s − 0.277·13-s + 0.438·15-s + 0.853·17-s − 0.856·19-s − 0.188·21-s − 1.61·23-s − 0.487·25-s − 0.994·27-s + 0.185·29-s + 1.17·31-s + 0.995·33-s − 0.219·35-s − 0.619·37-s − 0.169·39-s + 1.33·41-s + 1.96·43-s − 0.447·45-s + 1.66·47-s − 0.905·49-s + 0.522·51-s + 0.838·53-s + 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.837679171\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.837679171\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 - 1.60T + 5T^{2} \) |
| 7 | \( 1 + 0.812T + 7T^{2} \) |
| 11 | \( 1 - 5.39T + 11T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 + 7.75T + 23T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 - 8.55T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6.10T + 53T^{2} \) |
| 59 | \( 1 - 7.47T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 - 5.26T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 2.24T + 73T^{2} \) |
| 79 | \( 1 + 9.67T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 5.02T + 97T^{2} \) |
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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
−8.162505339282345710490072116477, −7.46458762271497884216937725217, −6.45514959062713846677335368801, −6.07390525612503519737959983964, −5.40503547007670400422489081147, −4.06512587813723036745326761039, −3.82311755206424671056035840296, −2.59179382035690188544224684882, −2.07669820378201200055141088732, −0.864626626947482912709667097409,
0.864626626947482912709667097409, 2.07669820378201200055141088732, 2.59179382035690188544224684882, 3.82311755206424671056035840296, 4.06512587813723036745326761039, 5.40503547007670400422489081147, 6.07390525612503519737959983964, 6.45514959062713846677335368801, 7.46458762271497884216937725217, 8.162505339282345710490072116477