| L(s) = 1 | + (1.19 − 0.758i)2-s + (−0.599 + 3.01i)3-s + (0.849 − 1.81i)4-s + (−1.78 − 1.19i)5-s + (1.57 + 4.05i)6-s + (1.99 − 0.825i)7-s + (−0.359 − 2.80i)8-s + (−5.95 − 2.46i)9-s + (−3.03 − 0.0695i)10-s + (−3.15 + 0.626i)11-s + (4.94 + 3.64i)12-s + (0.0943 − 0.0630i)13-s + (1.75 − 2.49i)14-s + (4.66 − 4.66i)15-s + (−2.55 − 3.07i)16-s + (2.42 + 2.42i)17-s + ⋯ |
| L(s) = 1 | + (0.843 − 0.536i)2-s + (−0.346 + 1.74i)3-s + (0.424 − 0.905i)4-s + (−0.798 − 0.533i)5-s + (0.641 + 1.65i)6-s + (0.753 − 0.311i)7-s + (−0.127 − 0.991i)8-s + (−1.98 − 0.822i)9-s + (−0.959 − 0.0219i)10-s + (−0.949 + 0.188i)11-s + (1.42 + 1.05i)12-s + (0.0261 − 0.0174i)13-s + (0.468 − 0.667i)14-s + (1.20 − 1.20i)15-s + (−0.639 − 0.768i)16-s + (0.587 + 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12441 + 0.0884835i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12441 + 0.0884835i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.19 + 0.758i)T \) |
| good | 3 | \( 1 + (0.599 - 3.01i)T + (-2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (1.78 + 1.19i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.99 + 0.825i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (3.15 - 0.626i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-0.0943 + 0.0630i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-2.42 - 2.42i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.93 - 2.89i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (1.33 - 3.22i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-2.01 - 0.401i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 4.16iT - 31T^{2} \) |
| 37 | \( 1 + (-5.48 + 8.20i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-0.347 + 0.839i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.925 - 4.65i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (8.31 + 8.31i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.565 + 0.112i)T + (48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (0.649 + 0.434i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (0.528 - 2.65i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-0.971 + 4.88i)T + (-61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-9.38 + 3.88i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-12.6 - 5.23i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (3.50 - 3.50i)T - 79iT^{2} \) |
| 83 | \( 1 + (8.25 + 12.3i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-4.98 - 12.0i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 4.06iT - 97T^{2} \) |
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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
−15.03275708893741029406675055066, −14.14842031311944412092415622375, −12.49471678783671229308285793835, −11.46293360762739200641384576547, −10.63382142818057126471641305252, −9.718900887898629418280226275970, −8.051172235354134488649996811693, −5.54804744515139580197013231643, −4.64068984854310141878158412900, −3.61682320104560084102309492155,
2.71604333004301834516227145896, 5.16436684260123782000689073814, 6.51120548984774388195918624719, 7.64479943838370142962658680926, 8.124347495348215441449389442923, 11.24143822679253081886187299597, 11.75146596030908303749914963216, 12.80765512018832278908391759494, 13.72262684916799241827988300315, 14.68507757976928866516500184756
