L(s) = 1 | + (−1.41 − 1.41i)5-s + 7-s + (−1.89 + 1.89i)11-s + (0.853 + 0.853i)13-s − 2.59i·17-s − 0.206i·23-s − 0.999i·25-s + (−3.10 + 3.10i)29-s − 1.70i·31-s + (−1.41 − 1.41i)35-s + (−1 + i)37-s − 11.0·41-s + (7.12 + 7.12i)43-s + 3.24·47-s + 49-s + ⋯ |
L(s) = 1 | + (−0.632 − 0.632i)5-s + 0.377·7-s + (−0.572 + 0.572i)11-s + (0.236 + 0.236i)13-s − 0.628i·17-s − 0.0431i·23-s − 0.199i·25-s + (−0.576 + 0.576i)29-s − 0.306i·31-s + (−0.239 − 0.239i)35-s + (−0.164 + 0.164i)37-s − 1.72·41-s + (1.08 + 1.08i)43-s + 0.472·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3635547844\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3635547844\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (1.41 + 1.41i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.89 - 1.89i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.853 - 0.853i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.59iT - 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 + 0.206iT - 23T^{2} \) |
| 29 | \( 1 + (3.10 - 3.10i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.70iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + (-7.12 - 7.12i)T + 43iT^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 + (0.722 + 0.722i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.00 - 5.00i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.14 + 3.14i)T + 61iT^{2} \) |
| 67 | \( 1 + (4.14 - 4.14i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.0iT - 71T^{2} \) |
| 73 | \( 1 + 5.66iT - 73T^{2} \) |
| 79 | \( 1 - 3.41iT - 79T^{2} \) |
| 83 | \( 1 + (-7.27 - 7.27i)T + 83iT^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.667454105662195791939917123912, −7.937650065089197043577717758117, −7.43093659267684727063953957166, −6.60461936358432782326033746532, −5.61121754028885563826577393693, −4.83630595611394473446830608206, −4.34690941192334445238943164823, −3.35277536190981476418637650682, −2.30597280907186412651803080779, −1.21665449077925040764236874292,
0.10846412090085402637824905712, 1.58551218791754511876558404843, 2.73183755871121786772755613528, 3.52280476773060428213073637344, 4.21236112033189944624198091600, 5.30128305884137017704312984015, 5.87268634461628294946316882598, 6.83920743837644002351933114089, 7.47507295396219633192558899716, 8.155670348812207404160186988785