| L(s) = 1 | − 5-s − 4.90·7-s − 11-s − 4.61·13-s − 3.76·17-s − 4.84·19-s − 0.860·23-s + 25-s + 10.3·29-s − 7.81·31-s + 4.90·35-s − 4.67·37-s − 2.97·41-s + 0.907·43-s − 13.2·47-s + 17.0·49-s − 5.13·53-s + 55-s − 12.5·59-s − 11.2·61-s + 4.61·65-s − 4.26·67-s − 5.13·71-s − 4.61·73-s + 4.90·77-s + 0.843·79-s − 5.75·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.85·7-s − 0.301·11-s − 1.27·13-s − 0.913·17-s − 1.11·19-s − 0.179·23-s + 0.200·25-s + 1.92·29-s − 1.40·31-s + 0.829·35-s − 0.768·37-s − 0.464·41-s + 0.138·43-s − 1.92·47-s + 2.44·49-s − 0.705·53-s + 0.134·55-s − 1.63·59-s − 1.43·61-s + 0.572·65-s − 0.521·67-s − 0.609·71-s − 0.539·73-s + 0.559·77-s + 0.0949·79-s − 0.631·83-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
| \(L(1)\) |
\(\approx\) |
\(0.1141012335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1141012335\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4.90T + 7T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 + 0.860T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 - 0.907T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 4.26T + 67T^{2} \) |
| 71 | \( 1 + 5.13T + 71T^{2} \) |
| 73 | \( 1 + 4.61T + 73T^{2} \) |
| 79 | \( 1 - 0.843T + 79T^{2} \) |
| 83 | \( 1 + 5.75T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 8.54T + 97T^{2} \) |
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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
−7.73584290762053654609589716749, −7.03121152729855342319534145657, −6.53022840940579389663057787443, −5.98681430391559353535463580326, −4.85946607691875725976684528400, −4.37409062204963602341092518192, −3.31484110619574963794576541981, −2.88567785856475583752232716901, −1.92034318223022552215075745893, −0.15431495087677390738709659390,
0.15431495087677390738709659390, 1.92034318223022552215075745893, 2.88567785856475583752232716901, 3.31484110619574963794576541981, 4.37409062204963602341092518192, 4.85946607691875725976684528400, 5.98681430391559353535463580326, 6.53022840940579389663057787443, 7.03121152729855342319534145657, 7.73584290762053654609589716749
