L(s) = 1 | + (−0.365 + 0.930i)2-s + (−0.123 + 1.64i)3-s + (−0.733 − 0.680i)4-s + (0.826 + 0.563i)5-s + (−1.48 − 0.716i)6-s + (0.733 + 0.680i)7-s + (0.900 − 0.433i)8-s + (−1.71 − 0.257i)9-s + (−0.826 + 0.563i)10-s + (1.21 − 1.12i)12-s + (−0.900 + 0.433i)14-s + (−1.03 + 1.29i)15-s + (0.0747 + 0.997i)16-s + (0.865 − 1.49i)18-s + (−0.222 − 0.974i)20-s + (−1.21 + 1.12i)21-s + ⋯ |
L(s) = 1 | + (−0.365 + 0.930i)2-s + (−0.123 + 1.64i)3-s + (−0.733 − 0.680i)4-s + (0.826 + 0.563i)5-s + (−1.48 − 0.716i)6-s + (0.733 + 0.680i)7-s + (0.900 − 0.433i)8-s + (−1.71 − 0.257i)9-s + (−0.826 + 0.563i)10-s + (1.21 − 1.12i)12-s + (−0.900 + 0.433i)14-s + (−1.03 + 1.29i)15-s + (0.0747 + 0.997i)16-s + (0.865 − 1.49i)18-s + (−0.222 − 0.974i)20-s + (−1.21 + 1.12i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9094926789\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9094926789\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.365 - 0.930i)T \) |
| 5 | \( 1 + (-0.826 - 0.563i)T \) |
| 7 | \( 1 + (-0.733 - 0.680i)T \) |
good | 3 | \( 1 + (0.123 - 1.64i)T + (-0.988 - 0.149i)T^{2} \) |
| 11 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 13 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.142 - 0.0440i)T + (0.826 - 0.563i)T^{2} \) |
| 29 | \( 1 + (0.425 + 1.86i)T + (-0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 41 | \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.658 + 1.67i)T + (-0.733 - 0.680i)T^{2} \) |
| 53 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 61 | \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.914 + 1.14i)T + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (1.88 + 0.284i)T + (0.955 + 0.294i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.31211639276166441173760172547, −9.734317595906468772761855897325, −9.059124116061184487108592940661, −8.376424164096780162389955836766, −7.26408386523432013986932352268, −6.00489098393875909495864211230, −5.56133147093104563124879788109, −4.73177254426291277956017210099, −3.77503124429605556725499817850, −2.23193911291551178461193038426,
1.15040363770417922607503913459, 1.74059881538931158605859433554, 2.89789893294290507181932985923, 4.47006032678799768929159287214, 5.43021739980087364515215940829, 6.54948151342709462068481617692, 7.49517461690339490427166336834, 8.118157232775885370668794348112, 8.900774716059262998736264634494, 9.793968427449279027090321180298