| L(s) = 1 | − 2.54i·3-s + (−1.30 + 1.81i)5-s + 0.780i·7-s − 3.45·9-s + 5.16·11-s − 3.34i·13-s + (4.60 + 3.32i)15-s − 6.62i·17-s − 6.40·19-s + 1.98·21-s − i·23-s + (−1.58 − 4.74i)25-s + 1.15i·27-s + 0.0877·29-s + 3.29·31-s + ⋯ |
| L(s) = 1 | − 1.46i·3-s + (−0.584 + 0.811i)5-s + 0.295i·7-s − 1.15·9-s + 1.55·11-s − 0.928i·13-s + (1.19 + 0.857i)15-s − 1.60i·17-s − 1.47·19-s + 0.432·21-s − 0.208i·23-s + (−0.316 − 0.948i)25-s + 0.223i·27-s + 0.0162·29-s + 0.591·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.557959 - 1.08992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.557959 - 1.08992i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.30 - 1.81i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 2.54iT - 3T^{2} \) |
| 7 | \( 1 - 0.780iT - 7T^{2} \) |
| 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 + 3.34iT - 13T^{2} \) |
| 17 | \( 1 + 6.62iT - 17T^{2} \) |
| 19 | \( 1 + 6.40T + 19T^{2} \) |
| 29 | \( 1 - 0.0877T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 + 4.68iT - 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 + 2.62iT - 43T^{2} \) |
| 47 | \( 1 - 5.13iT - 47T^{2} \) |
| 53 | \( 1 + 8.80iT - 53T^{2} \) |
| 59 | \( 1 + 2.98T + 59T^{2} \) |
| 61 | \( 1 + 2.05T + 61T^{2} \) |
| 67 | \( 1 + 3.60iT - 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 3.43T + 79T^{2} \) |
| 83 | \( 1 + 3.62iT - 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 0.427iT - 97T^{2} \) |
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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
−9.755488280964409017106744042238, −8.674375045554810318920637068811, −7.998207861798486995708769123865, −6.99044726288376854305368202984, −6.72898928530837606155268560711, −5.77822097748288975140876215917, −4.30426425432459687999381744985, −3.08618378843209880532477682880, −2.10804595147466163768345107327, −0.60553210691568671448900935269,
1.57501676325003354652614005464, 3.63088755850380672814281006515, 4.16389947476425594091212152775, 4.64589034915056079061429949060, 5.97917812305238099603429422023, 6.84146407768621400905324645361, 8.293480799818272274258981187485, 8.802825339056164590740239014399, 9.471610173602630030023751223996, 10.33245867760254836648453376657
