| L(s) = 1 | − 2.70i·2-s − 5.34·4-s + (2.17 + 0.539i)5-s + 9.04i·8-s + (1.46 − 5.87i)10-s − 2·11-s − 0.921i·13-s + 13.8·16-s − 1.07i·17-s + 3.07·19-s + (−11.5 − 2.87i)20-s + 5.41i·22-s − 2.34i·23-s + (4.41 + 2.34i)25-s − 2.49·26-s + ⋯ |
| L(s) = 1 | − 1.91i·2-s − 2.67·4-s + (0.970 + 0.241i)5-s + 3.19i·8-s + (0.461 − 1.85i)10-s − 0.603·11-s − 0.255i·13-s + 3.45·16-s − 0.261i·17-s + 0.706·19-s + (−2.59 − 0.643i)20-s + 1.15i·22-s − 0.487i·23-s + (0.883 + 0.468i)25-s − 0.489·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.445032416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445032416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.17 - 0.539i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.70iT - 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 0.921iT - 13T^{2} \) |
| 17 | \( 1 + 1.07iT - 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 + 2.34iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 6.52iT - 43T^{2} \) |
| 47 | \( 1 + 4.68iT - 47T^{2} \) |
| 53 | \( 1 + 3.75iT - 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 + 4.68iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 + 8.34T + 89T^{2} \) |
| 97 | \( 1 + 8.43iT - 97T^{2} \) |
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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
−9.106249191593358257131836845120, −8.229541061573364636728075844367, −7.24131397809029083386102488164, −5.83798104231212307709343358734, −5.27234852272548785067541696545, −4.35342739496451027615910363289, −3.31620368900512514513234200802, −2.56547696483063511763760074006, −1.82655398677392774121832209764, −0.56550848713064643420962290903,
1.22215213643024866436128561551, 2.96411714493426832030478141394, 4.32080834356785249897568255854, 4.99072062952936892057256174608, 5.76171205159086367588876404526, 6.24446035015574646704715732003, 7.09361789578612209256183694802, 7.85680425389959540558869070459, 8.447684142975010449749385034265, 9.387509458724433613477892492084
