L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 0.999·15-s + (−0.499 − 0.866i)21-s − 0.999·27-s − 29-s + (−0.5 + 0.866i)31-s + (0.499 + 0.866i)33-s + 0.999·35-s + (0.499 − 0.866i)45-s + (−0.499 − 0.866i)49-s + (0.5 − 0.866i)53-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 0.999·15-s + (−0.499 − 0.866i)21-s − 0.999·27-s − 29-s + (−0.5 + 0.866i)31-s + (0.499 + 0.866i)33-s + 0.999·35-s + (0.499 − 0.866i)45-s + (−0.499 − 0.866i)49-s + (0.5 − 0.866i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.171609900\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.171609900\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.56960542980599139432910063008, −9.887364876054883765997833013985, −8.825158967966344825043179155360, −7.75622842397281138326330756989, −7.19760192463088888829712604857, −6.49294543615988995286632276312, −5.28550417542179721972182787911, −3.92006594587286479499080429671, −2.71114177024210431121502880609, −1.67152980012138698261184570844,
1.96179931205958697876066291763, 3.15307568079583808449612955107, 4.43936632309463661566148667803, 5.37966868091882362727087198527, 5.85593767120683256892465347312, 7.65080337952761155978342878634, 8.469444121854547520638316484297, 9.063708239301009610618427229504, 9.691033409159790931732633059033, 10.81062538326327691669575886295