L(s) = 1 | + 2-s + 2·3-s + 4-s − 2·5-s + 2·6-s + 4·7-s + 8-s + 9-s − 2·10-s + 2·12-s + 6·13-s + 4·14-s − 4·15-s + 16-s + 4·17-s + 18-s − 19-s − 2·20-s + 8·21-s + 8·23-s + 2·24-s − 25-s + 6·26-s − 4·27-s + 4·28-s − 6·29-s − 4·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.577·12-s + 1.66·13-s + 1.06·14-s − 1.03·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s − 0.447·20-s + 1.74·21-s + 1.66·23-s + 0.408·24-s − 1/5·25-s + 1.17·26-s − 0.769·27-s + 0.755·28-s − 1.11·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.246433363\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.246433363\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.186559707877477832002128664152, −7.69018850511978445879890212926, −7.20059803040440767770430659493, −5.92548928025798202779883011472, −5.30500889959580484477620430224, −4.35514917060077024987946517017, −3.68390324917874655737759440144, −3.21746882703677109779472587407, −2.04027471543660968730594318930, −1.22317529651694379186629304432,
1.22317529651694379186629304432, 2.04027471543660968730594318930, 3.21746882703677109779472587407, 3.68390324917874655737759440144, 4.35514917060077024987946517017, 5.30500889959580484477620430224, 5.92548928025798202779883011472, 7.20059803040440767770430659493, 7.69018850511978445879890212926, 8.186559707877477832002128664152