L(s) = 1 | + 1.07·2-s + 2.50·3-s − 0.837·4-s + 2.70·6-s − 4.80·7-s − 3.05·8-s + 3.28·9-s − 2.54·11-s − 2.09·12-s − 4.01·13-s − 5.18·14-s − 1.62·16-s + 4.78·17-s + 3.54·18-s + 3.43·19-s − 12.0·21-s − 2.74·22-s + 4.75·23-s − 7.67·24-s − 4.33·26-s + 0.723·27-s + 4.02·28-s + 3.27·29-s + 0.120·31-s + 4.36·32-s − 6.39·33-s + 5.15·34-s + ⋯ |
L(s) = 1 | + 0.762·2-s + 1.44·3-s − 0.418·4-s + 1.10·6-s − 1.81·7-s − 1.08·8-s + 1.09·9-s − 0.768·11-s − 0.605·12-s − 1.11·13-s − 1.38·14-s − 0.406·16-s + 1.16·17-s + 0.835·18-s + 0.788·19-s − 2.62·21-s − 0.586·22-s + 0.991·23-s − 1.56·24-s − 0.849·26-s + 0.139·27-s + 0.760·28-s + 0.607·29-s + 0.0215·31-s + 0.771·32-s − 1.11·33-s + 0.884·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.753527016\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.753527016\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 1.07T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 - 4.75T + 23T^{2} \) |
| 29 | \( 1 - 3.27T + 29T^{2} \) |
| 31 | \( 1 - 0.120T + 31T^{2} \) |
| 37 | \( 1 - 5.16T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 + 0.519T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 0.643T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 1.34T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 - 4.79T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 - 7.95T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 - 5.96T + 97T^{2} \) |
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
−8.004152598081657051695364677035, −7.44081190172582236748257558713, −6.73333947770534575946163376906, −5.74218419185891136180732815669, −5.21238014931113972183003153442, −4.17233369206523753551374615503, −3.50293706479997303003027066678, −2.74094144437927154057666846245, −2.71272598232545356045124960749, −0.69336700354620242259947555636,
0.69336700354620242259947555636, 2.71272598232545356045124960749, 2.74094144437927154057666846245, 3.50293706479997303003027066678, 4.17233369206523753551374615503, 5.21238014931113972183003153442, 5.74218419185891136180732815669, 6.73333947770534575946163376906, 7.44081190172582236748257558713, 8.004152598081657051695364677035