L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 3.41i·11-s + (−4 + 4i)13-s − 1.00·14-s − 1.00·16-s + (3.41 − 3.41i)17-s − 2.82i·19-s + (2.41 + 2.41i)22-s + (−0.828 − 0.828i)23-s + 5.65i·26-s + (−0.707 + 0.707i)28-s + 4.82·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + 1.02i·11-s + (−1.10 + 1.10i)13-s − 0.267·14-s − 0.250·16-s + (0.828 − 0.828i)17-s − 0.648i·19-s + (0.514 + 0.514i)22-s + (−0.172 − 0.172i)23-s + 1.10i·26-s + (−0.133 + 0.133i)28-s + 0.896·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.191756996\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.191756996\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 - 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (4 - 4i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.41 + 3.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 + (0.828 + 0.828i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + (-2.24 - 2.24i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.82iT - 41T^{2} \) |
| 43 | \( 1 + (-8.07 + 8.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.757 + 0.757i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.41 + 5.41i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + (5.58 + 5.58i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.82iT - 71T^{2} \) |
| 73 | \( 1 + (-7.07 + 7.07i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.65iT - 79T^{2} \) |
| 83 | \( 1 + (-4.82 - 4.82i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 + (-7.41 - 7.41i)T + 97iT^{2} \) |
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
−8.703823732564818599949843278514, −7.57017194988745981861529337323, −6.99736594989227501228524020859, −6.36443404995698919683120781707, −5.14464916689989158402647429335, −4.68064805368551518299335524531, −3.91179107788775218385828811412, −2.74330350723390561401731630015, −2.12895179207047191317585245611, −0.71873921477873437996708461310,
0.963382300773776048038622673884, 2.67356263157165126108825506957, 3.17303382254405491354622127774, 4.23038619623491200332514704539, 5.09235137449360328006964745444, 5.96556388729189918560825686561, 6.21845187694929376322180975507, 7.40701075465569345137619375346, 8.107618497394323786690192501480, 8.421401744701951596377059128493