| L(s) = 1 | + 1.70·3-s + 5-s + 0.630·7-s − 0.0783·9-s + 0.630·11-s + 4.34·13-s + 1.70·15-s + 4.63·17-s + 1.70·19-s + 1.07·21-s − 1.70·23-s + 25-s − 5.26·27-s + 29-s − 1.70·31-s + 1.07·33-s + 0.630·35-s + 3.36·37-s + 7.41·39-s + 6.49·41-s − 6.38·43-s − 0.0783·45-s − 9.12·47-s − 6.60·49-s + 7.91·51-s + 5.23·53-s + 0.630·55-s + ⋯ |
| L(s) = 1 | + 0.986·3-s + 0.447·5-s + 0.238·7-s − 0.0261·9-s + 0.190·11-s + 1.20·13-s + 0.441·15-s + 1.12·17-s + 0.392·19-s + 0.235·21-s − 0.356·23-s + 0.200·25-s − 1.01·27-s + 0.185·29-s − 0.306·31-s + 0.187·33-s + 0.106·35-s + 0.553·37-s + 1.18·39-s + 1.01·41-s − 0.974·43-s − 0.0116·45-s − 1.33·47-s − 0.943·49-s + 1.10·51-s + 0.719·53-s + 0.0850·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.053122330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.053122330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 - 0.630T + 7T^{2} \) |
| 11 | \( 1 - 0.630T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 - 3.36T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 - 4.49T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 1.70T + 67T^{2} \) |
| 71 | \( 1 + 0.183T + 71T^{2} \) |
| 73 | \( 1 + 1.70T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 1.10T + 89T^{2} \) |
| 97 | \( 1 + 7.28T + 97T^{2} \) |
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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
−8.966131603569981658520805552129, −8.194935548767129636017255851348, −7.79204267117253421463204522860, −6.65618298473413621441339784845, −5.87700554214311308448134790945, −5.10177581321018459368165745389, −3.83390463871161936885802481536, −3.26519804280623153015824757637, −2.22112468119223086852835898276, −1.18399148874369869498598711929,
1.18399148874369869498598711929, 2.22112468119223086852835898276, 3.26519804280623153015824757637, 3.83390463871161936885802481536, 5.10177581321018459368165745389, 5.87700554214311308448134790945, 6.65618298473413621441339784845, 7.79204267117253421463204522860, 8.194935548767129636017255851348, 8.966131603569981658520805552129
