L(s) = 1 | + 5-s + 1.41i·7-s + 11-s − 13-s + 17-s − 19-s − 23-s + 1.41i·31-s + 1.41i·35-s − 1.41i·37-s + 41-s + 43-s − 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯ |
L(s) = 1 | + 5-s + 1.41i·7-s + 11-s − 13-s + 17-s − 19-s − 23-s + 1.41i·31-s + 1.41i·35-s − 1.41i·37-s + 41-s + 43-s − 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.248053139\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.248053139\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.867407862659683404900532480026, −9.194040615529656349970865681908, −8.641408709000047708409357887267, −7.54386169340081588467820835961, −6.46782998805953865523129560527, −5.80471549177089831131357871858, −5.16388001754828158897653412547, −3.89187089406014943455718454888, −2.54406568015478086719457527751, −1.83283734780885241631386477879,
1.25591108254819659868895857152, 2.47849886880256181301006945127, 3.89756949744959439524906263284, 4.53235417807861078972735299753, 5.83447815758222056735076990041, 6.43600279256790155076679622790, 7.41320994977026549169208484515, 8.044330708331787325220962429865, 9.394340823552955517760951048873, 9.780824532793101687100032429123