L(s) = 1 | + (−0.0591 − 1.41i)2-s + i·3-s + (−1.99 + 0.167i)4-s + (1.41 − 0.0591i)6-s − 1.33·7-s + (0.353 + 2.80i)8-s − 9-s − 2.94i·11-s + (−0.167 − 1.99i)12-s − 2.04i·13-s + (0.0788 + 1.88i)14-s + (3.94 − 0.665i)16-s − 3.61·17-s + (0.0591 + 1.41i)18-s − 5.35i·19-s + ⋯ |
L(s) = 1 | + (−0.0418 − 0.999i)2-s + 0.577i·3-s + (−0.996 + 0.0835i)4-s + (0.576 − 0.0241i)6-s − 0.504·7-s + (0.125 + 0.992i)8-s − 0.333·9-s − 0.887i·11-s + (−0.0482 − 0.575i)12-s − 0.566i·13-s + (0.0210 + 0.503i)14-s + (0.986 − 0.166i)16-s − 0.876·17-s + (0.0139 + 0.333i)18-s − 1.22i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0344790 - 0.549016i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0344790 - 0.549016i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0591 + 1.41i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 + 2.94iT - 11T^{2} \) |
| 13 | \( 1 + 2.04iT - 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + 5.35iT - 19T^{2} \) |
| 23 | \( 1 + 8.59T + 23T^{2} \) |
| 29 | \( 1 + 5.26iT - 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 - 6.55iT - 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 + 8.50iT - 43T^{2} \) |
| 47 | \( 1 + 9.97T + 47T^{2} \) |
| 53 | \( 1 + 6.12iT - 53T^{2} \) |
| 59 | \( 1 + 4.75iT - 59T^{2} \) |
| 61 | \( 1 - 8.51iT - 61T^{2} \) |
| 67 | \( 1 - 10.6iT - 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 1.52iT - 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−10.22965682670011392470931379475, −9.614799362542995035744283444743, −8.717890024922613517394568054342, −8.000896277054957833219209629844, −6.42548837105786042684694005484, −5.41302932772298192960490374392, −4.32635461534388414763965068026, −3.40620100389946468601347042296, −2.36429255178007999394237042538, −0.30117491900860591649816704431,
1.86624157414278988443447680090, 3.70117816696416890318393086448, 4.71385474323476541732573721728, 5.98909546428856439326551573949, 6.56021009594169738357370446305, 7.50644234695293638164573007250, 8.186443926358151579258020999605, 9.292968901131230607190080301592, 9.855235021353804671253363259006, 11.01611822891837146835584901940