L(s) = 1 | + 3-s − 2.13·5-s + 9-s − 0.298·11-s + 6.43·13-s − 2.13·15-s + 1.11·17-s − 1.01·19-s + 4.29·23-s − 0.441·25-s + 27-s − 0.422·29-s − 10.6·31-s − 0.298·33-s − 5.65·37-s + 6.43·39-s + 8.32·41-s + 7.09·43-s − 2.13·45-s + 8.11·47-s + 1.11·51-s − 3.44·53-s + 0.637·55-s − 1.01·57-s + 6.46·59-s + 5.83·61-s − 13.7·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.954·5-s + 0.333·9-s − 0.0900·11-s + 1.78·13-s − 0.551·15-s + 0.270·17-s − 0.233·19-s + 0.896·23-s − 0.0883·25-s + 0.192·27-s − 0.0784·29-s − 1.91·31-s − 0.0519·33-s − 0.929·37-s + 1.03·39-s + 1.30·41-s + 1.08·43-s − 0.318·45-s + 1.18·47-s + 0.156·51-s − 0.472·53-s + 0.0859·55-s − 0.135·57-s + 0.841·59-s + 0.747·61-s − 1.70·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.129066068\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.129066068\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.13T + 5T^{2} \) |
| 11 | \( 1 + 0.298T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 + 0.422T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 - 7.09T + 43T^{2} \) |
| 47 | \( 1 - 8.11T + 47T^{2} \) |
| 53 | \( 1 + 3.44T + 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 + 5.05T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 79 | \( 1 + 7.69T + 79T^{2} \) |
| 83 | \( 1 - 6.03T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 0.512T + 97T^{2} \) |
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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
−8.367959628257485995691456285341, −7.55871795443411600647530911056, −7.13682422938694111805838589608, −6.10163729902125365843986256353, −5.42206006898363675308524729429, −4.25182210017421464562837751458, −3.76311890154040227166049881939, −3.11201977133959435210368152699, −1.91816150153738173301086604779, −0.805241075556060165557432657410,
0.805241075556060165557432657410, 1.91816150153738173301086604779, 3.11201977133959435210368152699, 3.76311890154040227166049881939, 4.25182210017421464562837751458, 5.42206006898363675308524729429, 6.10163729902125365843986256353, 7.13682422938694111805838589608, 7.55871795443411600647530911056, 8.367959628257485995691456285341