L(s) = 1 | + 19·5-s − 13·7-s + 65·11-s + 56·13-s − 108·17-s + 58·19-s + 66·23-s + 236·25-s + 118·29-s + 145·31-s − 247·35-s − 190·37-s − 430·41-s + 530·43-s + 74·47-s − 174·49-s − 295·53-s + 1.23e3·55-s + 628·59-s − 360·61-s + 1.06e3·65-s + 146·67-s − 388·71-s + 753·73-s − 845·77-s − 1.13e3·79-s + 153·83-s + ⋯ |
L(s) = 1 | + 1.69·5-s − 0.701·7-s + 1.78·11-s + 1.19·13-s − 1.54·17-s + 0.700·19-s + 0.598·23-s + 1.88·25-s + 0.755·29-s + 0.840·31-s − 1.19·35-s − 0.844·37-s − 1.63·41-s + 1.87·43-s + 0.229·47-s − 0.507·49-s − 0.764·53-s + 3.02·55-s + 1.38·59-s − 0.755·61-s + 2.03·65-s + 0.266·67-s − 0.648·71-s + 1.20·73-s − 1.25·77-s − 1.61·79-s + 0.202·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.670232638\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.670232638\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 19 T + p^{3} T^{2} \) |
| 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 - 65 T + p^{3} T^{2} \) |
| 13 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 + 108 T + p^{3} T^{2} \) |
| 19 | \( 1 - 58 T + p^{3} T^{2} \) |
| 23 | \( 1 - 66 T + p^{3} T^{2} \) |
| 29 | \( 1 - 118 T + p^{3} T^{2} \) |
| 31 | \( 1 - 145 T + p^{3} T^{2} \) |
| 37 | \( 1 + 190 T + p^{3} T^{2} \) |
| 41 | \( 1 + 430 T + p^{3} T^{2} \) |
| 43 | \( 1 - 530 T + p^{3} T^{2} \) |
| 47 | \( 1 - 74 T + p^{3} T^{2} \) |
| 53 | \( 1 + 295 T + p^{3} T^{2} \) |
| 59 | \( 1 - 628 T + p^{3} T^{2} \) |
| 61 | \( 1 + 360 T + p^{3} T^{2} \) |
| 67 | \( 1 - 146 T + p^{3} T^{2} \) |
| 71 | \( 1 + 388 T + p^{3} T^{2} \) |
| 73 | \( 1 - 753 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1136 T + p^{3} T^{2} \) |
| 83 | \( 1 - 153 T + p^{3} T^{2} \) |
| 89 | \( 1 - 850 T + p^{3} T^{2} \) |
| 97 | \( 1 - 391 T + p^{3} T^{2} \) |
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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
−9.033936464578262910592181623842, −8.604748334429395226579392519507, −6.92340615967233758795484677955, −6.46799541146039516993650182654, −6.02302539524127508424227759394, −4.93967269673351061039653578987, −3.88194587914708935763836115498, −2.88747950312506327422092450673, −1.77004990760740655566398669894, −0.988913982413931403301173335224,
0.988913982413931403301173335224, 1.77004990760740655566398669894, 2.88747950312506327422092450673, 3.88194587914708935763836115498, 4.93967269673351061039653578987, 6.02302539524127508424227759394, 6.46799541146039516993650182654, 6.92340615967233758795484677955, 8.604748334429395226579392519507, 9.033936464578262910592181623842