L(s) = 1 | + (1.08 − 0.903i)2-s + (−1.00 − 1.00i)3-s + (0.368 − 1.96i)4-s + (−2.00 − 0.186i)6-s + (−0.707 + 0.707i)7-s + (−1.37 − 2.47i)8-s − 0.967i·9-s + 0.466i·11-s + (−2.35 + 1.60i)12-s + (2.66 − 2.66i)13-s + (−0.131 + 1.40i)14-s + (−3.72 − 1.45i)16-s + (−3.26 − 3.26i)17-s + (−0.874 − 1.05i)18-s − 6.88·19-s + ⋯ |
L(s) = 1 | + (0.769 − 0.638i)2-s + (−0.581 − 0.581i)3-s + (0.184 − 0.982i)4-s + (−0.819 − 0.0762i)6-s + (−0.267 + 0.267i)7-s + (−0.485 − 0.874i)8-s − 0.322i·9-s + 0.140i·11-s + (−0.679 + 0.464i)12-s + (0.738 − 0.738i)13-s + (−0.0350 + 0.376i)14-s + (−0.931 − 0.362i)16-s + (−0.791 − 0.791i)17-s + (−0.206 − 0.248i)18-s − 1.58·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0325131 - 1.40516i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0325131 - 1.40516i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.08 + 0.903i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.00 + 1.00i)T + 3iT^{2} \) |
| 11 | \( 1 - 0.466iT - 11T^{2} \) |
| 13 | \( 1 + (-2.66 + 2.66i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.26 + 3.26i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.88T + 19T^{2} \) |
| 23 | \( 1 + (-2.22 - 2.22i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.62iT - 29T^{2} \) |
| 31 | \( 1 + 3.30iT - 31T^{2} \) |
| 37 | \( 1 + (-7.25 - 7.25i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.58T + 41T^{2} \) |
| 43 | \( 1 + (1.82 + 1.82i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.36 - 2.36i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.71 + 7.71i)T - 53iT^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 + (-10.8 + 10.8i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.62iT - 71T^{2} \) |
| 73 | \( 1 + (2.29 - 2.29i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.91T + 79T^{2} \) |
| 83 | \( 1 + (8.69 + 8.69i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.89iT - 89T^{2} \) |
| 97 | \( 1 + (-6.04 - 6.04i)T + 97iT^{2} \) |
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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
−10.23363132145211702113259284184, −9.385452222881229267654494759355, −8.369797858512951003629706218270, −6.93177327082891966737495363914, −6.33445089464543876683636104056, −5.56305320974856381654830760388, −4.45352989689631014317352056014, −3.33315913176603393728848085973, −2.10870269254651401756054524400, −0.59511215365751618482602305355,
2.30878506738457833217487517376, 3.93024208848350978450778129986, 4.38423737873726287403233339219, 5.50551330229273836295371664110, 6.35277202566303956797229725955, 7.01944599364065309937091395938, 8.341019283427664905325971128731, 8.871173296731294515923364097892, 10.27419781988988470864290219844, 11.00300121415943588339516547160