| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s + 4·13-s + 5·17-s + 19-s + 6·21-s − 2·23-s − 9·27-s + 8·29-s − 10·31-s + 6·37-s − 12·39-s + 3·41-s − 4·43-s + 4·47-s − 3·49-s − 15·51-s − 6·53-s − 3·57-s − 8·59-s − 10·61-s − 12·63-s − 67-s + 6·69-s + 12·71-s + 3·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s + 1.10·13-s + 1.21·17-s + 0.229·19-s + 1.30·21-s − 0.417·23-s − 1.73·27-s + 1.48·29-s − 1.79·31-s + 0.986·37-s − 1.92·39-s + 0.468·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 2.10·51-s − 0.824·53-s − 0.397·57-s − 1.04·59-s − 1.28·61-s − 1.51·63-s − 0.122·67-s + 0.722·69-s + 1.42·71-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.090329866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.090329866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.50637748495717, −13.98006416316057, −13.41104852596659, −12.77724836462488, −12.42550173677734, −12.09084044829110, −11.34035841971704, −11.07238512011219, −10.45226229035095, −10.10310395884686, −9.444357128580833, −9.016876570893502, −7.989238211110554, −7.681293939594346, −6.848512681967154, −6.355400185925401, −6.038817571208441, −5.495018361940009, −4.931500062305684, −4.261244621832277, −3.562532447533440, −3.049693341329178, −1.842219947876820, −1.104724242217787, −0.4897580502975929,
0.4897580502975929, 1.104724242217787, 1.842219947876820, 3.049693341329178, 3.562532447533440, 4.261244621832277, 4.931500062305684, 5.495018361940009, 6.038817571208441, 6.355400185925401, 6.848512681967154, 7.681293939594346, 7.989238211110554, 9.016876570893502, 9.444357128580833, 10.10310395884686, 10.45226229035095, 11.07238512011219, 11.34035841971704, 12.09084044829110, 12.42550173677734, 12.77724836462488, 13.41104852596659, 13.98006416316057, 14.50637748495717
