L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.860 − 2.06i)5-s + 0.707·7-s − 1.00i·9-s + (1.79 − 1.79i)11-s + (3.86 − 3.86i)13-s + (−2.06 − 0.850i)15-s + 0.244i·17-s + (1.53 + 1.53i)19-s + (0.500 − 0.500i)21-s + 6.92·23-s + (−3.51 + 3.55i)25-s + (−0.707 − 0.707i)27-s + (4.89 + 4.89i)29-s − 7.60·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.384 − 0.922i)5-s + 0.267·7-s − 0.333i·9-s + (0.542 − 0.542i)11-s + (1.07 − 1.07i)13-s + (−0.533 − 0.219i)15-s + 0.0593i·17-s + (0.352 + 0.352i)19-s + (0.109 − 0.109i)21-s + 1.44·23-s + (−0.703 + 0.710i)25-s + (−0.136 − 0.136i)27-s + (0.909 + 0.909i)29-s − 1.36·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.027567858\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.027567858\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.860 + 2.06i)T \) |
good | 7 | \( 1 - 0.707T + 7T^{2} \) |
| 11 | \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.86 + 3.86i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.244iT - 17T^{2} \) |
| 19 | \( 1 + (-1.53 - 1.53i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + (-4.89 - 4.89i)T + 29iT^{2} \) |
| 31 | \( 1 + 7.60T + 31T^{2} \) |
| 37 | \( 1 + (8.47 + 8.47i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.12iT - 41T^{2} \) |
| 43 | \( 1 + (-0.684 - 0.684i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.47iT - 47T^{2} \) |
| 53 | \( 1 + (-1.47 - 1.47i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.86 + 5.86i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.0537 + 0.0537i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.85 + 7.85i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.08iT - 71T^{2} \) |
| 73 | \( 1 + 9.69T + 73T^{2} \) |
| 79 | \( 1 - 7.34T + 79T^{2} \) |
| 83 | \( 1 + (-6.80 + 6.80i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.07iT - 89T^{2} \) |
| 97 | \( 1 - 1.39iT - 97T^{2} \) |
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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
−8.757466151740398156683129719731, −8.378081719713642061189211822843, −7.55162980359097904212613116297, −6.71500500209837007918872068541, −5.63585634214878447515511334439, −5.05531226733350548984675466350, −3.75769142501636033420884905738, −3.24905211422917802846264304496, −1.63774459138033605896946865049, −0.77254748595619994972733581871,
1.50979384736841018544526438845, 2.70865732416701807487779056905, 3.63949168786324357421280304259, 4.31241746456828694379983765790, 5.29443100827509600943728397379, 6.60147588860291885147293851690, 6.90814095004150839327569194465, 7.897276768357719344939853846508, 8.746836242951017709057616710954, 9.332910195506612455499027535350