| L(s) = 1 | + 5-s − 4·11-s − 6·13-s − 2·17-s − 4·19-s + 8·23-s + 25-s + 6·29-s − 31-s + 2·37-s − 10·41-s + 4·43-s − 7·49-s − 10·53-s − 4·55-s − 12·59-s + 2·61-s − 6·65-s + 4·67-s + 2·73-s + 4·83-s − 2·85-s + 14·89-s − 4·95-s + 18·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.179·31-s + 0.328·37-s − 1.56·41-s + 0.609·43-s − 49-s − 1.37·53-s − 0.539·55-s − 1.56·59-s + 0.256·61-s − 0.744·65-s + 0.488·67-s + 0.234·73-s + 0.439·83-s − 0.216·85-s + 1.48·89-s − 0.410·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.26993950605316, −13.46825807059013, −13.16563128154643, −12.53203871517757, −12.47981370855940, −11.59384671026002, −11.11603486870872, −10.55101674778868, −10.21328858382345, −9.711843326915888, −9.117834449109211, −8.685652494846460, −8.020171980366045, −7.554329344793445, −7.032600830192208, −6.468347507445297, −6.002303334299714, −5.088564059094210, −4.820165115777549, −4.597163403289786, −3.336133244408300, −2.977886230392615, −2.268554112852310, −1.890085271351710, −0.7597240704076956, 0,
0.7597240704076956, 1.890085271351710, 2.268554112852310, 2.977886230392615, 3.336133244408300, 4.597163403289786, 4.820165115777549, 5.088564059094210, 6.002303334299714, 6.468347507445297, 7.032600830192208, 7.554329344793445, 8.020171980366045, 8.685652494846460, 9.117834449109211, 9.711843326915888, 10.21328858382345, 10.55101674778868, 11.11603486870872, 11.59384671026002, 12.47981370855940, 12.53203871517757, 13.16563128154643, 13.46825807059013, 14.26993950605316
