| L(s) = 1 | − 0.332·2-s + 1.45·3-s − 1.88·4-s − 1.44·5-s − 0.485·6-s + 1.29·8-s − 0.868·9-s + 0.480·10-s + 5.95·11-s − 2.75·12-s − 2.11·15-s + 3.34·16-s + 4.32·17-s + 0.288·18-s − 1.95·19-s + 2.73·20-s − 1.97·22-s − 0.540·23-s + 1.88·24-s − 2.91·25-s − 5.64·27-s + 7.15·29-s + 0.701·30-s + 6.10·31-s − 3.69·32-s + 8.69·33-s − 1.43·34-s + ⋯ |
| L(s) = 1 | − 0.235·2-s + 0.842·3-s − 0.944·4-s − 0.646·5-s − 0.198·6-s + 0.457·8-s − 0.289·9-s + 0.151·10-s + 1.79·11-s − 0.796·12-s − 0.544·15-s + 0.837·16-s + 1.04·17-s + 0.0680·18-s − 0.449·19-s + 0.610·20-s − 0.421·22-s − 0.112·23-s + 0.385·24-s − 0.582·25-s − 1.08·27-s + 1.32·29-s + 0.128·30-s + 1.09·31-s − 0.653·32-s + 1.51·33-s − 0.246·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.778777293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.778777293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.332T + 2T^{2} \) |
| 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 11 | \( 1 - 5.95T + 11T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 0.540T + 23T^{2} \) |
| 29 | \( 1 - 7.15T + 29T^{2} \) |
| 31 | \( 1 - 6.10T + 31T^{2} \) |
| 37 | \( 1 + 8.02T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 + 6.26T + 47T^{2} \) |
| 53 | \( 1 + 2.77T + 53T^{2} \) |
| 59 | \( 1 - 0.851T + 59T^{2} \) |
| 61 | \( 1 - 6.77T + 61T^{2} \) |
| 67 | \( 1 - 0.987T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 - 9.13T + 73T^{2} \) |
| 79 | \( 1 + 0.131T + 79T^{2} \) |
| 83 | \( 1 + 2.66T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 + 6.58T + 97T^{2} \) |
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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
−8.095168969367997439163109673475, −7.35289271177852887827464172301, −6.48626024015275064269941202172, −5.76597377277361948015561022598, −4.79606570429156073537951169841, −4.03175055071204203347190352958, −3.66362950595143092270895161421, −2.85289482625467464620377308156, −1.60996502254929146897306879729, −0.69435462640595069879018882139,
0.69435462640595069879018882139, 1.60996502254929146897306879729, 2.85289482625467464620377308156, 3.66362950595143092270895161421, 4.03175055071204203347190352958, 4.79606570429156073537951169841, 5.76597377277361948015561022598, 6.48626024015275064269941202172, 7.35289271177852887827464172301, 8.095168969367997439163109673475
