L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 2·11-s + 15-s − 2·17-s − 2·19-s + 21-s − 23-s + 25-s − 27-s − 10·29-s − 2·31-s + 2·33-s + 35-s − 2·37-s + 2·41-s − 4·43-s − 45-s + 49-s + 2·51-s + 8·53-s + 2·55-s + 2·57-s + 12·59-s − 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s − 0.485·17-s − 0.458·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.359·31-s + 0.348·33-s + 0.169·35-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s + 1.09·53-s + 0.269·55-s + 0.264·57-s + 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6137663495\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6137663495\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.39531864461260993016957252854, −7.18808494914454906645468106535, −6.23806795721116627266495008347, −5.67158311858517147070053795588, −4.98981851793234666815407902064, −4.16324550933233922182840733890, −3.58315082039732534516816915046, −2.58363578924290887773213892295, −1.70159318949345522905350080572, −0.37389219000422522579997619162,
0.37389219000422522579997619162, 1.70159318949345522905350080572, 2.58363578924290887773213892295, 3.58315082039732534516816915046, 4.16324550933233922182840733890, 4.98981851793234666815407902064, 5.67158311858517147070053795588, 6.23806795721116627266495008347, 7.18808494914454906645468106535, 7.39531864461260993016957252854