L(s) = 1 | + 3-s − 0.0932·5-s − 3.86·7-s + 9-s − 4.61·11-s − 0.601·13-s − 0.0932·15-s − 0.832·17-s − 3.67·19-s − 3.86·21-s − 6.30·23-s − 4.99·25-s + 27-s + 4.62·29-s − 4.74·31-s − 4.61·33-s + 0.360·35-s + 5.70·37-s − 0.601·39-s − 4.02·41-s + 3.85·43-s − 0.0932·45-s − 2.09·47-s + 7.91·49-s − 0.832·51-s + 7.76·53-s + 0.430·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.0417·5-s − 1.45·7-s + 0.333·9-s − 1.39·11-s − 0.166·13-s − 0.0240·15-s − 0.201·17-s − 0.843·19-s − 0.842·21-s − 1.31·23-s − 0.998·25-s + 0.192·27-s + 0.858·29-s − 0.851·31-s − 0.803·33-s + 0.0608·35-s + 0.937·37-s − 0.0963·39-s − 0.627·41-s + 0.587·43-s − 0.0139·45-s − 0.305·47-s + 1.13·49-s − 0.116·51-s + 1.06·53-s + 0.0580·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.029424698\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.029424698\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 0.0932T + 5T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 + 0.601T + 13T^{2} \) |
| 17 | \( 1 + 0.832T + 17T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 - 4.62T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 - 4.38T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 - 8.12T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 4.23T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
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
−8.024783529472893652431808151779, −7.14267751529092346503721624454, −6.48564352532684289837101186256, −5.85443589940186763298441598138, −5.06096151147215322593228054769, −4.03581442320935693473582513985, −3.54750590522834890252831900185, −2.54484615811875143001192751733, −2.17372481983245762869181118046, −0.44973376264797522626689594614,
0.44973376264797522626689594614, 2.17372481983245762869181118046, 2.54484615811875143001192751733, 3.54750590522834890252831900185, 4.03581442320935693473582513985, 5.06096151147215322593228054769, 5.85443589940186763298441598138, 6.48564352532684289837101186256, 7.14267751529092346503721624454, 8.024783529472893652431808151779