L(s) = 1 | + 3.08·3-s + 0.786·5-s + 4.29·7-s + 6.51·9-s − 1.21·11-s − 5.08·13-s + 2.42·15-s − 2.29·17-s + 13.2·21-s + 7.67·23-s − 4.38·25-s + 10.8·27-s + 0.489·29-s − 3.74·33-s + 3.38·35-s + 2·37-s − 15.6·39-s + 4.16·41-s + 12.9·43-s + 5.12·45-s − 5.80·47-s + 11.4·49-s − 7.08·51-s − 1.93·53-s − 0.954·55-s + 11.0·59-s − 5.38·61-s + ⋯ |
L(s) = 1 | + 1.78·3-s + 0.351·5-s + 1.62·7-s + 2.17·9-s − 0.365·11-s − 1.41·13-s + 0.626·15-s − 0.557·17-s + 2.89·21-s + 1.60·23-s − 0.876·25-s + 2.08·27-s + 0.0909·29-s − 0.651·33-s + 0.571·35-s + 0.328·37-s − 2.51·39-s + 0.650·41-s + 1.97·43-s + 0.763·45-s − 0.847·47-s + 1.63·49-s − 0.991·51-s − 0.266·53-s − 0.128·55-s + 1.44·59-s − 0.688·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.404913086\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.404913086\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 5 | \( 1 - 0.786T + 5T^{2} \) |
| 7 | \( 1 - 4.29T + 7T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 + 2.29T + 17T^{2} \) |
| 23 | \( 1 - 7.67T + 23T^{2} \) |
| 29 | \( 1 - 0.489T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 + 5.80T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 3.57T + 97T^{2} \) |
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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
−8.771140832337982584057219695471, −7.954752522836267472114514276928, −7.59529867721916100089831999368, −6.90067996798743231009844712740, −5.44467805818704243667061411443, −4.69801964113468180093391700113, −4.06445593777776457704242302360, −2.73061410508352222150756985357, −2.32979184644113736424478440793, −1.37216735895438119132187551130,
1.37216735895438119132187551130, 2.32979184644113736424478440793, 2.73061410508352222150756985357, 4.06445593777776457704242302360, 4.69801964113468180093391700113, 5.44467805818704243667061411443, 6.90067996798743231009844712740, 7.59529867721916100089831999368, 7.954752522836267472114514276928, 8.771140832337982584057219695471