| L(s) = 1 | + (−2 − i)5-s − i·7-s + 11-s + i·13-s + 3i·17-s + 4·19-s + 2i·23-s + (3 + 4i)25-s − 29-s − 6·31-s + (−1 + 2i)35-s + 2i·37-s + 10·41-s − 9i·47-s − 49-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s − 0.377i·7-s + 0.301·11-s + 0.277i·13-s + 0.727i·17-s + 0.917·19-s + 0.417i·23-s + (0.600 + 0.800i)25-s − 0.185·29-s − 1.07·31-s + (−0.169 + 0.338i)35-s + 0.328i·37-s + 1.56·41-s − 1.31i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440759232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440759232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 19iT - 97T^{2} \) |
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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
−8.823119122533229188572544757696, −8.040308744991618938276326281613, −7.40794528817469207355792099242, −6.71155379787613252365207795857, −5.64043587076705905959206969005, −4.86588770348150749722188174600, −3.89437474811101725891791234961, −3.43537292792601509275304793206, −1.91225921083793986166537986778, −0.71510675440840225733353349964,
0.823221150642784923997291249179, 2.40920181554364112947725556477, 3.24376977365893799593660754000, 4.09213803699924224900500794848, 5.00499987150392852932738005082, 5.88779591828114297773622021105, 6.76400509947195908060244412103, 7.57135906939469198153598330464, 7.975913224169062189596603923494, 9.141657462671296774904878903325
