| L(s) = 1 | + 2-s + 4-s + 3.80·5-s + 7-s + 8-s + 3.80·10-s + 11-s − 6.44·13-s + 14-s + 16-s + 3.80·17-s + 6.64·19-s + 3.80·20-s + 22-s + 0.842·23-s + 9.44·25-s − 6.44·26-s + 28-s − 8.44·29-s − 9.40·31-s + 32-s + 3.80·34-s + 3.80·35-s + 9.60·37-s + 6.64·38-s + 3.80·40-s − 3.80·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.69·5-s + 0.377·7-s + 0.353·8-s + 1.20·10-s + 0.301·11-s − 1.78·13-s + 0.267·14-s + 0.250·16-s + 0.921·17-s + 1.52·19-s + 0.849·20-s + 0.213·22-s + 0.175·23-s + 1.88·25-s − 1.26·26-s + 0.188·28-s − 1.56·29-s − 1.68·31-s + 0.176·32-s + 0.651·34-s + 0.642·35-s + 1.57·37-s + 1.07·38-s + 0.600·40-s − 0.593·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.615541701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.615541701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3.80T + 5T^{2} \) |
| 13 | \( 1 + 6.44T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 - 6.64T + 19T^{2} \) |
| 23 | \( 1 - 0.842T + 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 31 | \( 1 + 9.40T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 + 9.28T + 59T^{2} \) |
| 61 | \( 1 - 8.75T + 61T^{2} \) |
| 67 | \( 1 - 1.15T + 67T^{2} \) |
| 71 | \( 1 + 6.75T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 - 2.84T + 79T^{2} \) |
| 83 | \( 1 + 9.80T + 83T^{2} \) |
| 89 | \( 1 + 5.15T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−9.701522198182578327904401679827, −9.086431194279069426066051570268, −7.57757689610496881944711607465, −7.17395496757360296755173465586, −5.93753404362871426181355861379, −5.42194327190271186074624051041, −4.81467675683958722081657223521, −3.38001615177652481692593974125, −2.37487282676375360395858562462, −1.49587078039262481950061262573,
1.49587078039262481950061262573, 2.37487282676375360395858562462, 3.38001615177652481692593974125, 4.81467675683958722081657223521, 5.42194327190271186074624051041, 5.93753404362871426181355861379, 7.17395496757360296755173465586, 7.57757689610496881944711607465, 9.086431194279069426066051570268, 9.701522198182578327904401679827
