L(s) = 1 | + 0.618·2-s − 1.61·4-s − 5-s + 4.23·7-s − 2.23·8-s − 0.618·10-s + 0.236·11-s + 2.61·14-s + 1.85·16-s + 3.47·17-s + 4.23·19-s + 1.61·20-s + 0.145·22-s − 3.76·23-s + 25-s − 6.85·28-s + 7.47·29-s + 5.61·32-s + 2.14·34-s − 4.23·35-s + 3·37-s + 2.61·38-s + 2.23·40-s − 11.9·41-s − 6.23·43-s − 0.381·44-s − 2.32·46-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 0.809·4-s − 0.447·5-s + 1.60·7-s − 0.790·8-s − 0.195·10-s + 0.0711·11-s + 0.699·14-s + 0.463·16-s + 0.842·17-s + 0.971·19-s + 0.361·20-s + 0.0311·22-s − 0.784·23-s + 0.200·25-s − 1.29·28-s + 1.38·29-s + 0.993·32-s + 0.368·34-s − 0.716·35-s + 0.493·37-s + 0.424·38-s + 0.353·40-s − 1.86·41-s − 0.950·43-s − 0.0575·44-s − 0.342·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.340079912\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.340079912\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 0.708T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.100412614685106831801772146647, −7.34146137357177518073743782989, −6.39254436004683582813404802107, −5.41962670096729503568232385547, −5.06872513047092423659189625960, −4.40764626336363909480021793246, −3.69868048639036736282884966731, −2.91226375364514096501580991219, −1.66547421798741475107881250981, −0.76781584062975058329137140400,
0.76781584062975058329137140400, 1.66547421798741475107881250981, 2.91226375364514096501580991219, 3.69868048639036736282884966731, 4.40764626336363909480021793246, 5.06872513047092423659189625960, 5.41962670096729503568232385547, 6.39254436004683582813404802107, 7.34146137357177518073743782989, 8.100412614685106831801772146647