L(s) = 1 | − 102.·5-s + 512.·7-s − 2.41e3·11-s + 3.77e3i·13-s − 1.56e3·17-s + (−2.73e3 − 6.28e3i)19-s + 626.·23-s − 5.12e3·25-s + 1.57e4i·29-s + 2.61e4i·31-s − 5.25e4·35-s − 4.94e4i·37-s − 1.19e5i·41-s − 1.33e4·43-s + 1.86e5·47-s + ⋯ |
L(s) = 1 | − 0.819·5-s + 1.49·7-s − 1.81·11-s + 1.71i·13-s − 0.318·17-s + (−0.398 − 0.916i)19-s + 0.0514·23-s − 0.327·25-s + 0.643i·29-s + 0.879i·31-s − 1.22·35-s − 0.977i·37-s − 1.73i·41-s − 0.167·43-s + 1.79·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.916i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.398 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.016935816\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.016935816\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2.73e3 + 6.28e3i)T \) |
good | 5 | \( 1 + 102.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 512.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 2.41e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 3.77e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 1.56e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 626.T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.57e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.61e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 4.94e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.19e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.33e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.86e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.04e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.82e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.53e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 5.19e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.68e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.85e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.73e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.27e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 6.07e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.82e5iT - 8.32e11T^{2} \) |
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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
−9.087629195741339052954415294173, −8.529373727435374724291624764527, −7.55580671405639815669177465247, −7.11598763741344012882889290471, −5.58383594429893646218262117412, −4.71465716648521887217829650152, −4.11296033005248403178004673604, −2.58200177040045551487701198538, −1.72236689999462615822121354678, −0.26239215391494756346910061296,
0.72152059225821554595429207517, 2.09430235568085717821704356003, 3.11092782245147642375073121061, 4.35626429566158548068918532793, 5.13944053009857810579007527028, 5.92299122104712079644198648645, 7.59768015110059507578143165745, 7.994776792549016330022701615777, 8.303738757621816393528161233593, 9.938843267233061682708720566135