| L(s) = 1 | + 2.23·5-s + 2.82i·7-s + 5.23·11-s − 4.03i·13-s + 1.08i·17-s + 19-s − 7.40i·23-s + 5.00·25-s + 4.47·29-s − 4·31-s + 6.32i·35-s + 6.86i·37-s + 6·41-s − 8.48i·43-s − 8.48i·47-s + ⋯ |
| L(s) = 1 | + 0.999·5-s + 1.06i·7-s + 1.57·11-s − 1.11i·13-s + 0.262i·17-s + 0.229·19-s − 1.54i·23-s + 1.00·25-s + 0.830·29-s − 0.718·31-s + 1.06i·35-s + 1.12i·37-s + 0.937·41-s − 1.29i·43-s − 1.23i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.660363178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.660363178\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 4.03iT - 13T^{2} \) |
| 17 | \( 1 - 1.08iT - 17T^{2} \) |
| 23 | \( 1 + 7.40iT - 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 6.86iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 8.48iT - 47T^{2} \) |
| 53 | \( 1 + 6.86iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 - 1.62iT - 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 13.7iT - 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.24iT - 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 4.44iT - 97T^{2} \) |
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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
−8.792606218971714283050722402832, −8.069579658029823313658661572927, −6.80558204674359402432246583316, −6.37542475504175247922998024079, −5.61720434979377117181129255215, −4.99344719509033247573456383025, −3.88300468940899278559685657296, −2.85777714833811899859602441744, −2.09072664846655886821084517725, −0.981132731575372089649099360540,
1.14046987715621341880202707899, 1.74680409084504223997522879143, 3.08226696543770044992251890418, 4.05168022954242917602404537545, 4.60122954604181793397462238691, 5.76257317297614985744846457736, 6.38068225836891101856435117736, 7.07270377543629462578738096862, 7.64602872150550967506000147523, 8.891483144776324363877110942042
