| L(s) = 1 | + (0.719 + 1.21i)2-s + (1.09 − 0.397i)3-s + (−0.966 + 1.75i)4-s + (0.615 − 3.49i)5-s + (1.26 + 1.04i)6-s + (−3.53 + 2.04i)7-s + (−2.82 + 0.0826i)8-s + (−1.26 + 1.06i)9-s + (4.69 − 1.76i)10-s + (0.260 + 0.150i)11-s + (−0.358 + 2.29i)12-s + (0.546 − 1.50i)13-s + (−5.02 − 2.83i)14-s + (−0.715 − 4.05i)15-s + (−2.13 − 3.38i)16-s + (3.58 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (0.508 + 0.861i)2-s + (0.629 − 0.229i)3-s + (−0.483 + 0.875i)4-s + (0.275 − 1.56i)5-s + (0.517 + 0.425i)6-s + (−1.33 + 0.771i)7-s + (−0.999 + 0.0292i)8-s + (−0.421 + 0.354i)9-s + (1.48 − 0.557i)10-s + (0.0784 + 0.0452i)11-s + (−0.103 + 0.662i)12-s + (0.151 − 0.416i)13-s + (−1.34 − 0.758i)14-s + (−0.184 − 1.04i)15-s + (−0.533 − 0.845i)16-s + (0.869 + 0.729i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14701 + 0.428161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14701 + 0.428161i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.719 - 1.21i)T \) |
| 19 | \( 1 + (-3.34 + 2.78i)T \) |
| good | 3 | \( 1 + (-1.09 + 0.397i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.615 + 3.49i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (3.53 - 2.04i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.260 - 0.150i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.546 + 1.50i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.58 - 3.00i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-3.07 + 0.541i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.344 - 0.410i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.08 + 3.60i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.42iT - 37T^{2} \) |
| 41 | \( 1 + (-2.70 - 7.42i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (2.23 + 0.393i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.11 + 2.51i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (5.12 - 0.903i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (4.17 + 3.50i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.594 + 3.37i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.29 - 4.44i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.743 + 4.21i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (8.50 - 3.09i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (0.187 - 0.0681i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.9 + 7.44i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.26 + 6.21i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.90 + 4.64i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−14.65591897459051028535311561543, −13.24675476914950867243412815266, −13.02913322366406677382551923401, −11.99220109450569392163671957279, −9.512302588254682447738467448979, −8.831271141356605739980877242498, −7.84644368519052779066034951358, −6.08567306646823718360710993726, −5.10394514549717531621415418465, −3.16753618430076477141385167648,
2.98615094361244422230341521145, 3.57328981837449753534348154315, 5.99398505926708027924370677873, 7.13098473459938365714599023387, 9.330520990161486662473278838817, 10.01079756597665898986560298128, 10.97172985890367242781759493109, 12.20393530620569858432990108897, 13.67093772246635880439902754491, 14.12512968434177919362565009980
