L(s) = 1 | + (0.309 − 0.951i)2-s + (−1.97 + 1.43i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (0.755 + 2.32i)6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.920 − 2.83i)9-s + 0.999·10-s + (3.13 + 1.07i)11-s + 2.44·12-s + (0.310 − 0.955i)13-s + (−0.809 + 0.587i)14-s + (−1.97 − 1.43i)15-s + (0.309 + 0.951i)16-s + (0.301 + 0.926i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (−1.14 + 0.829i)3-s + (−0.404 − 0.293i)4-s + (0.138 + 0.425i)5-s + (0.308 + 0.949i)6-s + (−0.305 − 0.222i)7-s + (−0.286 + 0.207i)8-s + (0.306 − 0.944i)9-s + 0.316·10-s + (0.946 + 0.323i)11-s + 0.705·12-s + (0.0860 − 0.264i)13-s + (−0.216 + 0.157i)14-s + (−0.510 − 0.371i)15-s + (0.0772 + 0.237i)16-s + (0.0730 + 0.224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0941615 + 0.305841i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0941615 + 0.305841i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (-3.13 - 1.07i)T \) |
good | 3 | \( 1 + (1.97 - 1.43i)T + (0.927 - 2.85i)T^{2} \) |
| 13 | \( 1 + (-0.310 + 0.955i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.301 - 0.926i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.15 - 3.01i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 + (2.34 + 1.70i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.99 - 6.15i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (7.92 + 5.75i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.50 - 1.82i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 9.46T + 43T^{2} \) |
| 47 | \( 1 + (5.90 - 4.28i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.23 + 3.79i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.729 - 0.530i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.19 - 6.76i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 + (-0.682 - 2.10i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.57 - 2.59i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.94 + 12.1i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.76 - 8.51i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 5.28T + 89T^{2} \) |
| 97 | \( 1 + (2.90 - 8.95i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.53360650757946747317910792826, −10.20192508091582977359952044569, −9.404222072627866215033857612862, −8.289679359835287210416204574856, −6.82412171752676594662486653469, −6.09733390671107077427774693593, −5.23569515234690739077331983858, −4.16835501427329493432564594092, −3.52428235631741518880659179969, −1.79583689561980103589442118572,
0.17162509590706195118572070361, 1.76698259139970037818572488649, 3.68292129076765261166366422251, 4.86679213151737069163410683927, 5.73430250664443792916830907663, 6.49871349315970597327939580029, 6.93302770604095012306481897543, 8.158041534847591219264665320080, 8.956774127732916688090988198975, 9.879539099752671412812089734090