L(s) = 1 | + 3-s − 3·5-s + 7-s + 9-s + 11-s + 2·13-s − 3·15-s − 4·17-s + 21-s − 6·23-s + 4·25-s + 27-s − 4·29-s + 2·31-s + 33-s − 3·35-s − 11·37-s + 2·39-s + 10·41-s − 8·43-s − 3·45-s − 4·47-s − 6·49-s − 4·51-s − 3·53-s − 3·55-s + 2·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.774·15-s − 0.970·17-s + 0.218·21-s − 1.25·23-s + 4/5·25-s + 0.192·27-s − 0.742·29-s + 0.359·31-s + 0.174·33-s − 0.507·35-s − 1.80·37-s + 0.320·39-s + 1.56·41-s − 1.21·43-s − 0.447·45-s − 0.583·47-s − 6/7·49-s − 0.560·51-s − 0.412·53-s − 0.404·55-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.05183596361924, −13.69158537180607, −13.05210615203466, −12.53062388013559, −12.08652212095985, −11.58997918228018, −11.04216413967580, −10.89084847346359, −10.00411446690628, −9.582217460374566, −8.871589087861581, −8.526008019753965, −7.973776655912255, −7.764875718841543, −7.018118938710296, −6.574717448485672, −6.004952962277207, −5.117223778348972, −4.641706502342554, −4.045631574255353, −3.592445745560381, −3.239851400239052, −2.134085661014811, −1.844554714945671, −0.7969102768207951, 0,
0.7969102768207951, 1.844554714945671, 2.134085661014811, 3.239851400239052, 3.592445745560381, 4.045631574255353, 4.641706502342554, 5.117223778348972, 6.004952962277207, 6.574717448485672, 7.018118938710296, 7.764875718841543, 7.973776655912255, 8.526008019753965, 8.871589087861581, 9.582217460374566, 10.00411446690628, 10.89084847346359, 11.04216413967580, 11.58997918228018, 12.08652212095985, 12.53062388013559, 13.05210615203466, 13.69158537180607, 14.05183596361924