L(s) = 1 | + 3·3-s + 33.0·7-s + 9·9-s − 48.3·11-s − 60.3·13-s − 17.7·17-s + 130.·19-s + 99.2·21-s − 70.8·23-s + 27·27-s + 104.·29-s + 210.·31-s − 145.·33-s + 300.·37-s − 181.·39-s + 240.·41-s + 108·43-s + 278.·47-s + 750.·49-s − 53.3·51-s + 328.·53-s + 392.·57-s − 889.·59-s − 241.·61-s + 297.·63-s − 103.·67-s − 212.·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·7-s + 0.333·9-s − 1.32·11-s − 1.28·13-s − 0.253·17-s + 1.58·19-s + 1.03·21-s − 0.642·23-s + 0.192·27-s + 0.669·29-s + 1.21·31-s − 0.765·33-s + 1.33·37-s − 0.743·39-s + 0.914·41-s + 0.383·43-s + 0.865·47-s + 2.18·49-s − 0.146·51-s + 0.851·53-s + 0.912·57-s − 1.96·59-s − 0.506·61-s + 0.595·63-s − 0.189·67-s − 0.370·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.119990509\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.119990509\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 33.0T + 343T^{2} \) |
| 11 | \( 1 + 48.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 70.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 240.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108T + 7.95e4T^{2} \) |
| 47 | \( 1 - 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 889.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 241.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 103.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 277.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 274.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 366.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 57.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 203.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−9.372783420025994038417886062849, −8.340661919045081631091633034023, −7.68880235023874252580015271091, −7.42556424170616120194743333112, −5.82023630005094069514386690949, −4.90238083442971768066309319949, −4.44863420620650766132696366503, −2.84867373886782349622594371706, −2.19081336113120710667779492294, −0.902478242542447817355648283085,
0.902478242542447817355648283085, 2.19081336113120710667779492294, 2.84867373886782349622594371706, 4.44863420620650766132696366503, 4.90238083442971768066309319949, 5.82023630005094069514386690949, 7.42556424170616120194743333112, 7.68880235023874252580015271091, 8.340661919045081631091633034023, 9.372783420025994038417886062849