| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 7-s + 8-s + 9-s + 2·10-s + 4·11-s + 12-s + 6·13-s + 14-s + 2·15-s + 16-s + 18-s − 4·19-s + 2·20-s + 21-s + 4·22-s − 8·23-s + 24-s − 25-s + 6·26-s + 27-s + 28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.743358571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.743358571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.17482492578561, −15.82817640807894, −14.92457128569613, −14.62371660069492, −13.99074981156007, −13.64902106390237, −13.15626920164496, −12.53296815797602, −11.74411104213080, −11.34923132284782, −10.59036397324645, −10.00904027778041, −9.379186515874091, −8.666529979269971, −8.258017251344012, −7.451481070609276, −6.562122681996199, −5.998571916303095, −5.845839916401697, −4.518297642967953, −4.104624060544993, −3.502794051372474, −2.465177605941952, −1.812560144023453, −1.141357092012553,
1.141357092012553, 1.812560144023453, 2.465177605941952, 3.502794051372474, 4.104624060544993, 4.518297642967953, 5.845839916401697, 5.998571916303095, 6.562122681996199, 7.451481070609276, 8.258017251344012, 8.666529979269971, 9.379186515874091, 10.00904027778041, 10.59036397324645, 11.34923132284782, 11.74411104213080, 12.53296815797602, 13.15626920164496, 13.64902106390237, 13.99074981156007, 14.62371660069492, 14.92457128569613, 15.82817640807894, 16.17482492578561
