| L(s) = 1 | + (1.45 − 0.308i)2-s + (0.266 − 1.71i)3-s + (0.183 − 0.0816i)4-s + (0.0707 + 0.0150i)5-s + (−0.141 − 2.56i)6-s + (−0.203 + 1.93i)7-s + (−2.15 + 1.56i)8-s + (−2.85 − 0.910i)9-s + 0.107·10-s + (3.25 − 0.610i)11-s + (−0.0909 − 0.335i)12-s + (0.742 + 0.824i)13-s + (0.302 + 2.87i)14-s + (0.0445 − 0.117i)15-s + (−2.91 + 3.24i)16-s + (−0.0576 − 0.177i)17-s + ⋯ |
| L(s) = 1 | + (1.02 − 0.218i)2-s + (0.153 − 0.988i)3-s + (0.0917 − 0.0408i)4-s + (0.0316 + 0.00672i)5-s + (−0.0578 − 1.04i)6-s + (−0.0770 + 0.733i)7-s + (−0.763 + 0.554i)8-s + (−0.952 − 0.303i)9-s + 0.0339·10-s + (0.982 − 0.184i)11-s + (−0.0262 − 0.0969i)12-s + (0.205 + 0.228i)13-s + (0.0808 + 0.768i)14-s + (0.0115 − 0.0302i)15-s + (−0.729 + 0.810i)16-s + (−0.0139 − 0.0430i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39757 - 0.542769i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39757 - 0.542769i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.266 + 1.71i)T \) |
| 11 | \( 1 + (-3.25 + 0.610i)T \) |
| good | 2 | \( 1 + (-1.45 + 0.308i)T + (1.82 - 0.813i)T^{2} \) |
| 5 | \( 1 + (-0.0707 - 0.0150i)T + (4.56 + 2.03i)T^{2} \) |
| 7 | \( 1 + (0.203 - 1.93i)T + (-6.84 - 1.45i)T^{2} \) |
| 13 | \( 1 + (-0.742 - 0.824i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.0576 + 0.177i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.20 - 3.05i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.42 + 5.93i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.337 + 3.21i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-1.10 - 1.22i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (7.75 + 5.63i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.529 + 5.03i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-4.06 - 7.03i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.93 - 3.53i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-0.467 + 1.43i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (4.84 - 2.15i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-4.73 + 5.25i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (0.447 - 0.775i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.476 + 1.46i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-9.31 - 6.76i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-12.9 + 2.75i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (12.0 - 13.3i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + (10.2 - 2.17i)T + (88.6 - 39.4i)T^{2} \) |
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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
−13.83477923313632387265675589476, −12.51747215729735313616841130063, −12.26636574704058522273915956850, −11.11206099395363919736297342486, −9.112915551398739159497847938486, −8.335264723472617141762522004534, −6.58394127969582390352964999751, −5.71774802218106025235724905372, −3.99119036820923900059262934875, −2.38969404137463023082822904052,
3.48143470626888574839013394029, 4.37333417218524324796483209685, 5.58449427182586789881642411274, 6.93848584597814154352320717803, 8.789878770175484267687754017713, 9.703446287103939137305197888588, 10.87826191677207213308181085090, 12.02586248925259006465444676448, 13.41483349619565344057689023745, 13.96415593366434024133923328183
