L(s) = 1 | − 2-s + 3·3-s − 4-s − 2·5-s − 3·6-s − 2·7-s + 3·8-s + 6·9-s + 2·10-s − 11-s − 3·12-s + 4·13-s + 2·14-s − 6·15-s − 16-s + 17-s − 6·18-s + 8·19-s + 2·20-s − 6·21-s + 22-s + 4·23-s + 9·24-s − 25-s − 4·26-s + 9·27-s + 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.894·5-s − 1.22·6-s − 0.755·7-s + 1.06·8-s + 2·9-s + 0.632·10-s − 0.301·11-s − 0.866·12-s + 1.10·13-s + 0.534·14-s − 1.54·15-s − 1/4·16-s + 0.242·17-s − 1.41·18-s + 1.83·19-s + 0.447·20-s − 1.30·21-s + 0.213·22-s + 0.834·23-s + 1.83·24-s − 1/5·25-s − 0.784·26-s + 1.73·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.944063584\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.944063584\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.102869052627858211859139789732, −7.33965752462953423279688270655, −7.06687180105908810888988659981, −5.73336762321120738194283244513, −4.81138054303974873329907854689, −3.82594739249932802240463786115, −3.54507916061554421451149285926, −2.88269598038921643679091862129, −1.65721031369962887800848770765, −0.74476572811836920933469460867,
0.74476572811836920933469460867, 1.65721031369962887800848770765, 2.88269598038921643679091862129, 3.54507916061554421451149285926, 3.82594739249932802240463786115, 4.81138054303974873329907854689, 5.73336762321120738194283244513, 7.06687180105908810888988659981, 7.33965752462953423279688270655, 8.102869052627858211859139789732