L(s) = 1 | + i·2-s − 4-s + (−2 + i)5-s − 2i·7-s − i·8-s + (−1 − 2i)10-s − 2·11-s − i·13-s + 2·14-s + 16-s + 2i·17-s + 4·19-s + (2 − i)20-s − 2i·22-s + (3 − 4i)25-s + 26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.755i·7-s − 0.353i·8-s + (−0.316 − 0.632i)10-s − 0.603·11-s − 0.277i·13-s + 0.534·14-s + 0.250·16-s + 0.485i·17-s + 0.917·19-s + (0.447 − 0.223i)20-s − 0.426i·22-s + (0.600 − 0.800i)25-s + 0.196·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.178112087\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178112087\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−9.979970515295386925813278834114, −8.744488276021039587369213120821, −7.977949416082132357355638043698, −7.42680077058700292471204770823, −6.70138623465424849284267852003, −5.69199562201546320598422739492, −4.63486277788420801135174057666, −3.84389128685193229933124713034, −2.81451474276305788264398277562, −0.72864149967538039579016075059,
0.930918142397487302835950278504, 2.50507709629848732940770638817, 3.34827504757729037250009898440, 4.56485444308437704603388022604, 5.12909958999602973102193775545, 6.28144202217683831298153005352, 7.53615272390464044425244229702, 8.171815759576235563455702877760, 8.997031072783284288748607379997, 9.654174481263098400305920372138