| L(s) = 1 | − 7-s + 2·11-s + 6·13-s + 8·17-s + 19-s + 2·23-s − 5·25-s − 6·29-s + 10·37-s + 8·41-s − 12·43-s − 4·47-s + 49-s − 2·53-s + 8·59-s − 10·61-s − 12·67-s + 10·71-s − 2·73-s − 2·77-s − 8·79-s + 4·83-s + 8·89-s − 6·91-s + 2·97-s + 4·101-s − 2·107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s + 1.66·13-s + 1.94·17-s + 0.229·19-s + 0.417·23-s − 25-s − 1.11·29-s + 1.64·37-s + 1.24·41-s − 1.82·43-s − 0.583·47-s + 1/7·49-s − 0.274·53-s + 1.04·59-s − 1.28·61-s − 1.46·67-s + 1.18·71-s − 0.234·73-s − 0.227·77-s − 0.900·79-s + 0.439·83-s + 0.847·89-s − 0.628·91-s + 0.203·97-s + 0.398·101-s − 0.193·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.276681053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.276681053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.141126835189670259468431790625, −7.71179363869623773766951421262, −6.78425966044936463996649660556, −5.92461069948544809607605082020, −5.68657546334957466916060281266, −4.45128064822373452531412119614, −3.58432542369787507380741066676, −3.17785935881446270719128297999, −1.73108969966905643937793181595, −0.895071874805042573229366117238,
0.895071874805042573229366117238, 1.73108969966905643937793181595, 3.17785935881446270719128297999, 3.58432542369787507380741066676, 4.45128064822373452531412119614, 5.68657546334957466916060281266, 5.92461069948544809607605082020, 6.78425966044936463996649660556, 7.71179363869623773766951421262, 8.141126835189670259468431790625
