L(s) = 1 | + (1.41 + 1.73i)5-s + (2.44 − i)7-s − 3·9-s + 3.46·11-s − 3.46i·13-s + 4·17-s − 2.82i·19-s − 4.89·23-s + (−0.999 + 4.89i)25-s − 6.92i·29-s + 9.79·31-s + (5.19 + 2.82i)35-s + 5.65·37-s − 9.79i·41-s − 2.82i·43-s + ⋯ |
L(s) = 1 | + (0.632 + 0.774i)5-s + (0.925 − 0.377i)7-s − 9-s + 1.04·11-s − 0.960i·13-s + 0.970·17-s − 0.648i·19-s − 1.02·23-s + (−0.199 + 0.979i)25-s − 1.28i·29-s + 1.75·31-s + (0.878 + 0.478i)35-s + 0.929·37-s − 1.53i·41-s − 0.431i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.273196370\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.273196370\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.41 - 1.73i)T \) |
| 7 | \( 1 + (-2.44 + i)T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 9.79T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + 9.79iT - 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 - 10iT - 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 8.48iT - 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 - 12T + 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−9.049870365645307062366327617320, −8.013672026063980509538947220990, −7.70721141685149559907167923819, −6.46399012652705304060626452898, −5.98796664750650594446923636726, −5.14403262590952117720177862655, −4.08575456872885243882454651757, −3.08714535313144076249349072865, −2.22861624234354253888874488207, −0.919918536663050212440372084544,
1.22070792569754039858983283889, 1.97798408771781646243855333806, 3.23856992150028245952581665077, 4.44142861658886955459727317879, 5.01899863753516906468121556874, 6.05560722701594332748530384943, 6.36209149966536343450516703970, 7.87897204563158970083659564803, 8.262732345113026709979488234681, 9.131970749828879777903803695386