| L(s) = 1 | − 0.399·5-s − 9.64·7-s + 9.57·11-s + 9.40i·13-s − 25.3·17-s + (1.88 + 18.9i)19-s + 14.2·23-s − 24.8·25-s + 10.9i·29-s − 12.8i·31-s + 3.85·35-s + 3.29i·37-s − 63.6i·41-s + 26.8·43-s + 18.1·47-s + ⋯ |
| L(s) = 1 | − 0.0799·5-s − 1.37·7-s + 0.870·11-s + 0.723i·13-s − 1.49·17-s + (0.0989 + 0.995i)19-s + 0.621·23-s − 0.993·25-s + 0.377i·29-s − 0.415i·31-s + 0.110·35-s + 0.0890i·37-s − 1.55i·41-s + 0.625·43-s + 0.386·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0989 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0989 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7907791023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7907791023\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.88 - 18.9i)T \) |
| good | 5 | \( 1 + 0.399T + 25T^{2} \) |
| 7 | \( 1 + 9.64T + 49T^{2} \) |
| 11 | \( 1 - 9.57T + 121T^{2} \) |
| 13 | \( 1 - 9.40iT - 169T^{2} \) |
| 17 | \( 1 + 25.3T + 289T^{2} \) |
| 23 | \( 1 - 14.2T + 529T^{2} \) |
| 29 | \( 1 - 10.9iT - 841T^{2} \) |
| 31 | \( 1 + 12.8iT - 961T^{2} \) |
| 37 | \( 1 - 3.29iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 63.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 26.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 18.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 36.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 12.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 71.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 36.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 63.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 120. iT - 9.40e3T^{2} \) |
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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
−8.781867720736052763107464583568, −7.55990620305272341077283447095, −6.81848842820817245811159533627, −6.32309129786168182445686532521, −5.54240281890802756634058869660, −4.20180637161080790749755596929, −3.83220195592453968317338535707, −2.72096041475222273349925406032, −1.68193614713807994801197847575, −0.23564365144164671560746182245,
0.814114202080014157251476945571, 2.31286964395138782960002902207, 3.16480949834070636402376295280, 3.98416926365993534763768050698, 4.86319602945568383739294831717, 5.95145941048622465411909259102, 6.58593209012244686770606934856, 7.08806531198423903078820808962, 8.128326210998533376465394704123, 9.017389292537943383067939294821
