L(s) = 1 | − 3.18·3-s + 0.345·5-s − 2.57·7-s + 7.15·9-s − 2.58·11-s + 5.15·13-s − 1.10·15-s + 7.10·17-s − 19-s + 8.18·21-s + 4.50·23-s − 4.88·25-s − 13.2·27-s − 3.13·29-s + 7.52·31-s + 8.24·33-s − 0.888·35-s − 2.97·37-s − 16.4·39-s + 11.7·41-s + 5.64·43-s + 2.47·45-s + 4.74·47-s − 0.393·49-s − 22.6·51-s − 8.86·53-s − 0.895·55-s + ⋯ |
L(s) = 1 | − 1.83·3-s + 0.154·5-s − 0.971·7-s + 2.38·9-s − 0.780·11-s + 1.42·13-s − 0.284·15-s + 1.72·17-s − 0.229·19-s + 1.78·21-s + 0.939·23-s − 0.976·25-s − 2.54·27-s − 0.582·29-s + 1.35·31-s + 1.43·33-s − 0.150·35-s − 0.489·37-s − 2.62·39-s + 1.84·41-s + 0.860·43-s + 0.368·45-s + 0.691·47-s − 0.0562·49-s − 3.16·51-s − 1.21·53-s − 0.120·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9110767212\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9110767212\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 - 0.345T + 5T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 23 | \( 1 - 4.50T + 23T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 + 2.97T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 + 8.86T + 53T^{2} \) |
| 59 | \( 1 - 3.42T + 59T^{2} \) |
| 61 | \( 1 - 4.45T + 61T^{2} \) |
| 67 | \( 1 + 5.30T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 1.31T + 73T^{2} \) |
| 83 | \( 1 + 8.97T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5.46T + 97T^{2} \) |
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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
−7.80125055242163565471498572796, −7.25027028891365879117235333703, −6.35472986841985566946288814392, −5.86083061471691184162701443008, −5.58376677068266196630242016655, −4.59325026290539472111473402196, −3.79106784835543745928219868644, −2.90010947097350737824672353646, −1.39738821553590756044254559038, −0.60955003567654261805559887435,
0.60955003567654261805559887435, 1.39738821553590756044254559038, 2.90010947097350737824672353646, 3.79106784835543745928219868644, 4.59325026290539472111473402196, 5.58376677068266196630242016655, 5.86083061471691184162701443008, 6.35472986841985566946288814392, 7.25027028891365879117235333703, 7.80125055242163565471498572796