| L(s) = 1 | + 3i·3-s + (10 − 5i)5-s + 10i·7-s − 9·9-s − 46·11-s + 34i·13-s + (15 + 30i)15-s − 66i·17-s − 104·19-s − 30·21-s − 164i·23-s + (75 − 100i)25-s − 27i·27-s + 224·29-s + 72·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.894 − 0.447i)5-s + 0.539i·7-s − 0.333·9-s − 1.26·11-s + 0.725i·13-s + (0.258 + 0.516i)15-s − 0.941i·17-s − 1.25·19-s − 0.311·21-s − 1.48i·23-s + (0.599 − 0.800i)25-s − 0.192i·27-s + 1.43·29-s + 0.417·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.537722138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537722138\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 + (-10 + 5i)T \) |
| good | 7 | \( 1 - 10iT - 343T^{2} \) |
| 11 | \( 1 + 46T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 104T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 224T + 2.43e4T^{2} \) |
| 31 | \( 1 - 72T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 194T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 480iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 286iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 426T + 2.05e5T^{2} \) |
| 61 | \( 1 + 698T + 2.26e5T^{2} \) |
| 67 | \( 1 + 328iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 188T + 3.57e5T^{2} \) |
| 73 | \( 1 + 740iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 412iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3iT - 9.12e5T^{2} \) |
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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
−9.467437670454232112234383501244, −8.766644010844473602937549864723, −8.126768815843957617172444397341, −6.72185089074116046775563195391, −5.98711442494287936637141828110, −4.95904737334328059198075661642, −4.49983328596882906459509705342, −2.80072094046420710867696922456, −2.15704981035420949541059276785, −0.39508833229165773712208172791,
1.14301097533282647415889360878, 2.30957186702099164991215068515, 3.17605593430784549350289147412, 4.59009741789531277580639603400, 5.73895342183115544031714318363, 6.24791699268170186524000313511, 7.34321019391167641477339616812, 7.958821920559701306408864216731, 8.883251106493685934255539400981, 10.05718288977837683351891995446
