L(s) = 1 | − 2.18·3-s − 3.62·5-s − 4.31·7-s + 1.79·9-s + 11-s − 6.10·13-s + 7.94·15-s − 0.694·17-s + 19-s + 9.43·21-s − 7.79·23-s + 8.16·25-s + 2.64·27-s + 6.02·29-s − 6.26·31-s − 2.18·33-s + 15.6·35-s + 10.1·37-s + 13.3·39-s − 2.04·41-s − 2.16·43-s − 6.49·45-s − 10.2·47-s + 11.5·49-s + 1.52·51-s − 1.95·53-s − 3.62·55-s + ⋯ |
L(s) = 1 | − 1.26·3-s − 1.62·5-s − 1.62·7-s + 0.596·9-s + 0.301·11-s − 1.69·13-s + 2.05·15-s − 0.168·17-s + 0.229·19-s + 2.05·21-s − 1.62·23-s + 1.63·25-s + 0.509·27-s + 1.11·29-s − 1.12·31-s − 0.380·33-s + 2.64·35-s + 1.66·37-s + 2.13·39-s − 0.319·41-s − 0.330·43-s − 0.968·45-s − 1.48·47-s + 1.65·49-s + 0.212·51-s − 0.268·53-s − 0.489·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1569732565\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1569732565\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 17 | \( 1 + 0.694T + 17T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 2.04T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 - 6.84T + 61T^{2} \) |
| 67 | \( 1 + 8.35T + 67T^{2} \) |
| 71 | \( 1 - 0.566T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 7.05T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 2.89T + 97T^{2} \) |
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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
−10.13949295348326475154839395824, −9.706361494556107217129911128626, −8.400897897517294561979692376938, −7.39441993315294767120578597129, −6.76215227232990532972608667821, −5.94059986326449947047341869855, −4.79387166570874318850564394816, −3.94694090848552384518281460721, −2.86951455601453425008270805189, −0.31729387406430854674398202656,
0.31729387406430854674398202656, 2.86951455601453425008270805189, 3.94694090848552384518281460721, 4.79387166570874318850564394816, 5.94059986326449947047341869855, 6.76215227232990532972608667821, 7.39441993315294767120578597129, 8.400897897517294561979692376938, 9.706361494556107217129911128626, 10.13949295348326475154839395824