L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s − 2·7-s + 8-s + 9-s + 2·10-s − 11-s + 12-s + 13-s − 2·14-s + 2·15-s + 16-s + 18-s + 5·19-s + 2·20-s − 2·21-s − 22-s − 23-s + 24-s − 25-s + 26-s + 27-s − 2·28-s − 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 − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s + 1.14·19-s + 0.447·20-s − 0.436·21-s − 0.213·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.400741764\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.400741764\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.53327295654091, −15.40936501871599, −14.33844716754905, −13.87971929747465, −13.79624415835771, −12.99892328531133, −12.73334009788635, −11.95048001627301, −11.44542098884824, −10.65140250571550, −9.984147206504061, −9.690156211814175, −9.178426109162802, −8.132809279752338, −7.938785523644383, −6.883514943261424, −6.494104808328103, −5.933626976425978, −5.162382945708329, −4.695832705604949, −3.595795331195336, −3.292509817878215, −2.474093292314062, −1.855676888929896, −0.8393406836487740,
0.8393406836487740, 1.855676888929896, 2.474093292314062, 3.292509817878215, 3.595795331195336, 4.695832705604949, 5.162382945708329, 5.933626976425978, 6.494104808328103, 6.883514943261424, 7.938785523644383, 8.132809279752338, 9.178426109162802, 9.690156211814175, 9.984147206504061, 10.65140250571550, 11.44542098884824, 11.95048001627301, 12.73334009788635, 12.99892328531133, 13.79624415835771, 13.87971929747465, 14.33844716754905, 15.40936501871599, 15.53327295654091