| L(s) = 1 | − 0.115i·3-s − 1.52·5-s + 8.98·9-s − 4.17i·11-s + 8.98·13-s + 0.176i·15-s + 21.6·17-s − 26.6i·19-s + 11.4i·23-s − 22.6·25-s − 2.08i·27-s − 31.6·29-s + 9.10i·31-s − 0.483·33-s + 1.02·37-s + ⋯ |
| L(s) = 1 | − 0.0385i·3-s − 0.304·5-s + 0.998·9-s − 0.379i·11-s + 0.691·13-s + 0.0117i·15-s + 1.27·17-s − 1.40i·19-s + 0.499i·23-s − 0.907·25-s − 0.0771i·27-s − 1.09·29-s + 0.293i·31-s − 0.0146·33-s + 0.0277·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.063263969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.063263969\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.115iT - 9T^{2} \) |
| 5 | \( 1 + 1.52T + 25T^{2} \) |
| 11 | \( 1 + 4.17iT - 121T^{2} \) |
| 13 | \( 1 - 8.98T + 169T^{2} \) |
| 17 | \( 1 - 21.6T + 289T^{2} \) |
| 19 | \( 1 + 26.6iT - 361T^{2} \) |
| 23 | \( 1 - 11.4iT - 529T^{2} \) |
| 29 | \( 1 + 31.6T + 841T^{2} \) |
| 31 | \( 1 - 9.10iT - 961T^{2} \) |
| 37 | \( 1 - 1.02T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 35.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 4.01iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 79.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 71.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 69.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 25.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 125. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 33.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 8.98T + 9.40e3T^{2} \) |
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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
−9.241208678634059448539919415496, −8.253083390359505060212225808545, −7.54957763314061454531292270764, −6.84218442549633128581936618659, −5.86237222516330868951130684384, −5.00500079163511693981385665427, −3.95583534692726402967389595060, −3.25481103250109113844278881707, −1.82695044901071254221747926836, −0.68683491892878704477999872415,
1.06029860310480422841191834835, 2.09952992063649657576148441267, 3.68401546836073886454470142959, 4.00475554551239750582111179535, 5.30612696744396148383430127641, 6.01992570591983142838892856660, 7.10041379021348704617177082201, 7.71197080843502130105794261404, 8.432429738053857018276307706224, 9.486157893062379501158915223205
