L(s) = 1 | + 5-s + 7-s − 11-s + 2·13-s + 3·17-s + 19-s + 6·23-s + 25-s + 9·29-s − 5·31-s + 35-s + 5·37-s + 6·41-s − 8·43-s + 6·47-s − 6·49-s − 9·53-s − 55-s + 6·59-s + 5·61-s + 2·65-s − 8·67-s − 9·71-s − 10·73-s − 77-s − 14·79-s − 6·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s + 0.727·17-s + 0.229·19-s + 1.25·23-s + 1/5·25-s + 1.67·29-s − 0.898·31-s + 0.169·35-s + 0.821·37-s + 0.937·41-s − 1.21·43-s + 0.875·47-s − 6/7·49-s − 1.23·53-s − 0.134·55-s + 0.781·59-s + 0.640·61-s + 0.248·65-s − 0.977·67-s − 1.06·71-s − 1.17·73-s − 0.113·77-s − 1.57·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.668652681\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.668652681\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−7.76224759039196068845633021230, −7.25157086883659729226587891101, −6.36034346747155479569436187655, −5.80130996456210235966979083538, −5.01746829440136474739842547657, −4.46265397762780440267206695762, −3.34784116424407914210834671768, −2.77661197900664329111792555619, −1.67264343971143917086525034509, −0.862693104681156284849854332348,
0.862693104681156284849854332348, 1.67264343971143917086525034509, 2.77661197900664329111792555619, 3.34784116424407914210834671768, 4.46265397762780440267206695762, 5.01746829440136474739842547657, 5.80130996456210235966979083538, 6.36034346747155479569436187655, 7.25157086883659729226587891101, 7.76224759039196068845633021230