| L(s) = 1 | + (1.24 − 0.673i)2-s + (0.0518 + 1.73i)3-s + (1.09 − 1.67i)4-s + (1.55 + 1.84i)5-s + (1.23 + 2.11i)6-s + (−1.49 + 0.544i)7-s + (0.231 − 2.81i)8-s + (−2.99 + 0.179i)9-s + (3.17 + 1.25i)10-s + (−0.102 + 0.122i)11-s + (2.95 + 1.80i)12-s + (3.39 − 0.597i)13-s + (−1.49 + 1.68i)14-s + (−3.12 + 2.78i)15-s + (−1.61 − 3.66i)16-s + (−1.43 − 2.48i)17-s + ⋯ |
| L(s) = 1 | + (0.879 − 0.476i)2-s + (0.0299 + 0.999i)3-s + (0.546 − 0.837i)4-s + (0.693 + 0.826i)5-s + (0.502 + 0.864i)6-s + (−0.565 + 0.205i)7-s + (0.0817 − 0.996i)8-s + (−0.998 + 0.0597i)9-s + (1.00 + 0.396i)10-s + (−0.0308 + 0.0367i)11-s + (0.853 + 0.521i)12-s + (0.940 − 0.165i)13-s + (−0.399 + 0.450i)14-s + (−0.805 + 0.718i)15-s + (−0.402 − 0.915i)16-s + (−0.348 − 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98171 + 0.298241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98171 + 0.298241i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 0.673i)T \) |
| 3 | \( 1 + (-0.0518 - 1.73i)T \) |
| good | 5 | \( 1 + (-1.55 - 1.84i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.49 - 0.544i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.102 - 0.122i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.39 + 0.597i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.43 + 2.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.75 + 2.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.14 - 0.781i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-8.60 - 1.51i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (6.69 + 2.43i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (5.14 - 2.97i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.74 + 9.90i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.59 - 9.05i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.66 + 1.33i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 9.80iT - 53T^{2} \) |
| 59 | \( 1 + (-1.72 - 2.05i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.492 + 1.35i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.225 + 0.0397i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.82 - 4.88i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.14 + 12.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.18 - 12.3i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-11.1 - 1.95i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (3.73 - 6.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.17 - 4.34i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.37167269514584702516815186183, −11.13317332775561649162785855778, −10.62639812152095762281929444352, −9.778221631896865136568455997986, −8.809013700082969397079119795015, −6.75865102868778297988981869217, −6.00226336412011008176044115285, −4.84880023122399940069931329267, −3.50669132139941559161663830537, −2.54814644185125516427538358639,
1.84448699220802643999315152356, 3.53487302134121306703463587222, 5.11999355828027466363555814959, 6.19735188495666924782403583450, 6.80050525478674444767263377899, 8.257906772489456074748152773646, 8.860606333185815299198246096670, 10.54951294448426034152547209800, 11.73488948120645073836129490727, 12.78014295565693483184367266964
