L(s) = 1 | + 1.75·3-s − 5-s + 0.706·7-s + 0.0928·9-s − 0.467·11-s − 0.643·13-s − 1.75·15-s + 5.29·17-s + 4.59·19-s + 1.24·21-s − 8.36·23-s + 25-s − 5.11·27-s + 0.263·29-s + 2.56·31-s − 0.822·33-s − 0.706·35-s + 9.10·37-s − 1.13·39-s + 4.62·41-s − 3.61·43-s − 0.0928·45-s − 5.16·47-s − 6.50·49-s + 9.31·51-s + 11.0·53-s + 0.467·55-s + ⋯ |
L(s) = 1 | + 1.01·3-s − 0.447·5-s + 0.267·7-s + 0.0309·9-s − 0.141·11-s − 0.178·13-s − 0.454·15-s + 1.28·17-s + 1.05·19-s + 0.271·21-s − 1.74·23-s + 0.200·25-s − 0.983·27-s + 0.0489·29-s + 0.460·31-s − 0.143·33-s − 0.119·35-s + 1.49·37-s − 0.181·39-s + 0.722·41-s − 0.551·43-s − 0.0138·45-s − 0.753·47-s − 0.928·49-s + 1.30·51-s + 1.51·53-s + 0.0631·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.722975172\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.722975172\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 1.75T + 3T^{2} \) |
| 7 | \( 1 - 0.706T + 7T^{2} \) |
| 11 | \( 1 + 0.467T + 11T^{2} \) |
| 13 | \( 1 + 0.643T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 4.59T + 19T^{2} \) |
| 23 | \( 1 + 8.36T + 23T^{2} \) |
| 29 | \( 1 - 0.263T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 - 9.10T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 + 3.61T + 43T^{2} \) |
| 47 | \( 1 + 5.16T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 - 6.45T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 - 0.301T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 9.95T + 97T^{2} \) |
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
−7.85824628326983632802653866589, −7.58094952683296560555660346258, −6.48435964579294106627579965208, −5.71088510300547701019423815223, −5.01914500250426959162961803214, −4.05219302325013612832846412788, −3.46957670164297083208863511683, −2.75779832800197554079556465183, −1.94208527380319129303731593970, −0.77560042546034993827727201609,
0.77560042546034993827727201609, 1.94208527380319129303731593970, 2.75779832800197554079556465183, 3.46957670164297083208863511683, 4.05219302325013612832846412788, 5.01914500250426959162961803214, 5.71088510300547701019423815223, 6.48435964579294106627579965208, 7.58094952683296560555660346258, 7.85824628326983632802653866589