| L(s) = 1 | + 1.99·2-s − 3-s + 1.96·4-s − 3.98·5-s − 1.99·6-s − 7-s − 0.0694·8-s + 9-s − 7.94·10-s − 3.40·11-s − 1.96·12-s + 1.09·13-s − 1.99·14-s + 3.98·15-s − 4.06·16-s − 5.53·17-s + 1.99·18-s + 4.31·19-s − 7.83·20-s + 21-s − 6.78·22-s − 8.56·23-s + 0.0694·24-s + 10.9·25-s + 2.17·26-s − 27-s − 1.96·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 0.577·3-s + 0.982·4-s − 1.78·5-s − 0.812·6-s − 0.377·7-s − 0.0245·8-s + 0.333·9-s − 2.51·10-s − 1.02·11-s − 0.567·12-s + 0.302·13-s − 0.532·14-s + 1.02·15-s − 1.01·16-s − 1.34·17-s + 0.469·18-s + 0.989·19-s − 1.75·20-s + 0.218·21-s − 1.44·22-s − 1.78·23-s + 0.0141·24-s + 2.18·25-s + 0.426·26-s − 0.192·27-s − 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5334589205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5334589205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 1.99T + 2T^{2} \) |
| 5 | \( 1 + 3.98T + 5T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 + 8.56T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 + 7.94T + 37T^{2} \) |
| 41 | \( 1 + 7.62T + 41T^{2} \) |
| 43 | \( 1 + 1.15T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 0.990T + 61T^{2} \) |
| 67 | \( 1 + 5.82T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 1.48T + 73T^{2} \) |
| 79 | \( 1 - 6.76T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 8.07T + 97T^{2} \) |
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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
−7.36732448838728545843799811124, −7.21854702448880331484691304216, −6.27916930287089350472549104097, −5.51520989754085406889634311669, −5.01894466917773242384759076405, −4.09669258094005513963590724080, −3.86378814633320701363496623674, −3.09511021709549650127317364610, −2.09622723518732165748514191201, −0.27866664237060272025617432132,
0.27866664237060272025617432132, 2.09622723518732165748514191201, 3.09511021709549650127317364610, 3.86378814633320701363496623674, 4.09669258094005513963590724080, 5.01894466917773242384759076405, 5.51520989754085406889634311669, 6.27916930287089350472549104097, 7.21854702448880331484691304216, 7.36732448838728545843799811124
