L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 6·11-s + 15-s − 6·17-s − 19-s − 2·21-s + 8·23-s + 25-s − 27-s + 4·29-s − 6·33-s − 2·35-s + 4·37-s + 2·43-s − 45-s + 8·47-s − 3·49-s + 6·51-s + 2·53-s − 6·55-s + 57-s − 12·59-s + 2·61-s + 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.258·15-s − 1.45·17-s − 0.229·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.04·33-s − 0.338·35-s + 0.657·37-s + 0.304·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.840·51-s + 0.274·53-s − 0.809·55-s + 0.132·57-s − 1.56·59-s + 0.256·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.787928889\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.787928889\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.454214734883251473999014766804, −7.46307212019793678556314757234, −6.76242068169946844350861892331, −6.35568709931231127456553997796, −5.30054231386447702035951774158, −4.42342128541001589497390333987, −4.15312467491244982662604138290, −2.92380519859586694297776399100, −1.70399671261848221618934256367, −0.818960836178036513910564281349,
0.818960836178036513910564281349, 1.70399671261848221618934256367, 2.92380519859586694297776399100, 4.15312467491244982662604138290, 4.42342128541001589497390333987, 5.30054231386447702035951774158, 6.35568709931231127456553997796, 6.76242068169946844350861892331, 7.46307212019793678556314757234, 8.454214734883251473999014766804