| L(s) = 1 | + (1.13 + 0.838i)2-s + (−0.601 − 1.18i)3-s + (0.593 + 1.90i)4-s + (−2.23 − 0.138i)5-s + (0.305 − 1.84i)6-s + (0.707 + 0.707i)7-s + (−0.925 + 2.67i)8-s + (0.731 − 1.00i)9-s + (−2.42 − 2.02i)10-s + (1.48 + 2.04i)11-s + (1.89 − 1.84i)12-s + (0.469 + 2.96i)13-s + (0.212 + 1.39i)14-s + (1.17 + 2.71i)15-s + (−3.29 + 2.26i)16-s + (3.39 + 1.73i)17-s + ⋯ |
| L(s) = 1 | + (0.805 + 0.592i)2-s + (−0.347 − 0.681i)3-s + (0.296 + 0.954i)4-s + (−0.998 − 0.0620i)5-s + (0.124 − 0.754i)6-s + (0.267 + 0.267i)7-s + (−0.327 + 0.944i)8-s + (0.243 − 0.335i)9-s + (−0.766 − 0.641i)10-s + (0.448 + 0.617i)11-s + (0.547 − 0.533i)12-s + (0.130 + 0.822i)13-s + (0.0567 + 0.373i)14-s + (0.304 + 0.701i)15-s + (−0.823 + 0.566i)16-s + (0.824 + 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39038 + 1.17374i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39038 + 1.17374i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.13 - 0.838i)T \) |
| 5 | \( 1 + (2.23 + 0.138i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| good | 3 | \( 1 + (0.601 + 1.18i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-1.48 - 2.04i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.469 - 2.96i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-3.39 - 1.73i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.631 - 1.94i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.787 - 4.96i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-7.44 - 2.41i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.98 - 0.970i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.27 + 0.994i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (7.63 + 5.54i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (8.51 - 8.51i)T - 43iT^{2} \) |
| 47 | \( 1 + (-11.4 + 5.82i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (3.45 - 1.76i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (8.24 + 5.98i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.05 + 5.85i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.561 - 1.10i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-1.80 - 0.587i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-12.0 - 1.91i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-1.09 + 3.35i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (7.27 + 3.70i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (6.01 + 8.28i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.16 - 8.17i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.09417448281840122363779759890, −9.664617624867657906870856739403, −8.552645866086177344453790685586, −7.74742969541054695806444479963, −7.03237882445223359441038196006, −6.35728434287613097292164483329, −5.24654127361535249620223089517, −4.20875659147191979890114286655, −3.40982049819813084244404526333, −1.62645712560606940210121294497,
0.846576487405344930517988434356, 2.83861028617848460212051563905, 3.81904100202857004572737059021, 4.60726197721955996878349145310, 5.37229696752317861074653920980, 6.52093367018808652434163756660, 7.59727557772405703733861831091, 8.556381877178455353549225381556, 9.836659291791050994405423288847, 10.50785229599129476564482964787
