| L(s) = 1 | − 3.40i·3-s + 2.23i·5-s − 8.62·9-s − 3.62·11-s + 1.06i·13-s + 7.62·15-s + 5.75i·17-s − 5.00·25-s + 19.1i·27-s − 9.62·29-s + 12.3i·33-s + 3.62·39-s − 19.2i·45-s − 1.28i·47-s + 19.6·51-s + ⋯ |
| L(s) = 1 | − 1.96i·3-s + 0.999i·5-s − 2.87·9-s − 1.09·11-s + 0.294i·13-s + 1.96·15-s + 1.39i·17-s − 1.00·25-s + 3.68i·27-s − 1.78·29-s + 2.15i·33-s + 0.580·39-s − 2.87i·45-s − 0.187i·47-s + 2.74·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.209677 + 0.209677i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.209677 + 0.209677i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.40iT - 3T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 13 | \( 1 - 1.06iT - 13T^{2} \) |
| 17 | \( 1 - 5.75iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 9.62T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 1.28iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 19.3iT - 97T^{2} \) |
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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
−10.52987244295761349942071747197, −9.177352883822216329254034165072, −8.087291376810580175745950863155, −7.68606348320110424573478910580, −6.88805447018201216643203686908, −6.15544116708940073903249860969, −5.46894980965681602556743159239, −3.57108196342745047868552116707, −2.50489303706815910617302787264, −1.72315154565527323611574176674,
0.12637045002210007206319343398, 2.59940965940910232599333700771, 3.64531185440243123138512574419, 4.61237974804640420487341340096, 5.23121996784486624516601132760, 5.78159341043706604398730403175, 7.56318404100237797361711986361, 8.422742458816043206909153787378, 9.220026303187029576364604276801, 9.677299819623877435097101991405
