| L(s) = 1 | − i·2-s + 7·4-s + 6i·7-s − 15i·8-s + 43·11-s + 28i·13-s + 6·14-s + 41·16-s − 91i·17-s + 35·19-s − 43i·22-s + 162i·23-s + 28·26-s + 42i·28-s + 160·29-s + ⋯ |
| L(s) = 1 | − 0.353i·2-s + 0.875·4-s + 0.323i·7-s − 0.662i·8-s + 1.17·11-s + 0.597i·13-s + 0.114·14-s + 0.640·16-s − 1.29i·17-s + 0.422·19-s − 0.416i·22-s + 1.46i·23-s + 0.211·26-s + 0.283i·28-s + 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.31773 - 0.547143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31773 - 0.547143i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + iT - 8T^{2} \) |
| 7 | \( 1 - 6iT - 343T^{2} \) |
| 11 | \( 1 - 43T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 91iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 35T + 6.85e3T^{2} \) |
| 23 | \( 1 - 162iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 160T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42T + 2.97e4T^{2} \) |
| 37 | \( 1 + 314iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 203T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 196iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 82iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 280T + 2.05e5T^{2} \) |
| 61 | \( 1 + 518T + 2.26e5T^{2} \) |
| 67 | \( 1 - 141iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 412T + 3.57e5T^{2} \) |
| 73 | \( 1 - 763iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 510T + 4.93e5T^{2} \) |
| 83 | \( 1 - 777iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 945T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.24e3iT - 9.12e5T^{2} \) |
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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
−11.78095794529568528931988329931, −11.01358803702866093824695124426, −9.737666004305842109144221759153, −9.045275310979518361918703915143, −7.49758683097558234352435081383, −6.73218584991688115356403756762, −5.57983661243711358920732504523, −3.97636953211741518269571920036, −2.67550728019049928456824006338, −1.26783699447599961263571658716,
1.33272796765119163125405994647, 2.96998460118144349797316446203, 4.42126816620113659008484445807, 6.01895959926985832442616390006, 6.64904153775204110324849651449, 7.81631347999061760717321019399, 8.703886115521945247123125525885, 10.13650849674616674161009520178, 10.84989625072399154783647284807, 11.91024676504293148752800662682
