| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 8-s + 9-s + 10-s − 4·11-s + 2·12-s + 2·13-s + 2·15-s + 16-s + 8·17-s + 18-s − 6·19-s + 20-s − 4·22-s − 4·23-s + 2·24-s + 25-s + 2·26-s − 4·27-s − 6·29-s + 2·30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.577·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.94·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.852·22-s − 0.834·23-s + 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 1.11·29-s + 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.071649458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.071649458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| 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
−10.78524842246065370835052081258, −10.18076940563670721337529296568, −9.102066376137223109803910203953, −8.118957846682560780166935032553, −7.54815335398458076593938530060, −6.10181652811109734263548661708, −5.34673915974432534760216274251, −3.93281458879405714793180621994, −2.99518962250547346483426263862, −1.97892037150738648722925117261,
1.97892037150738648722925117261, 2.99518962250547346483426263862, 3.93281458879405714793180621994, 5.34673915974432534760216274251, 6.10181652811109734263548661708, 7.54815335398458076593938530060, 8.118957846682560780166935032553, 9.102066376137223109803910203953, 10.18076940563670721337529296568, 10.78524842246065370835052081258
