| L(s) = 1 | + (−1 − 2i)5-s + 3·9-s + 4i·13-s − 4i·17-s + 4·19-s − 8i·23-s + (−3 + 4i)25-s − 2·29-s + 8·31-s − 8i·37-s − 6·41-s − 8i·43-s + (−3 − 6i)45-s − 8i·47-s − 4·59-s + ⋯ |
| L(s) = 1 | + (−0.447 − 0.894i)5-s + 9-s + 1.10i·13-s − 0.970i·17-s + 0.917·19-s − 1.66i·23-s + (−0.600 + 0.800i)25-s − 0.371·29-s + 1.43·31-s − 1.31i·37-s − 0.937·41-s − 1.21i·43-s + (−0.447 − 0.894i)45-s − 1.16i·47-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28325 - 0.793098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28325 - 0.793098i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−9.718041539332464626372441339389, −9.064975301251896972903238645405, −8.257272721551026313630727910791, −7.26938529790489518941486150390, −6.66267580602216179562967404843, −5.27094420487273348104079897818, −4.55558651605682777273395501259, −3.75797069881104069637612385180, −2.17543261314394075770418611888, −0.795800109119008986207509747722,
1.38123482634235965320934201771, 2.97490547179958909482585564847, 3.71245667115707378202967247638, 4.85222644296536460126174026308, 5.97249057170617375311967569198, 6.82286378000839713123443864296, 7.73245463829029371021455980558, 8.142734267157461736751636478283, 9.664454064208399134561197183584, 10.05233075025307437457255620931
