| L(s) = 1 | − 0.892·3-s − 2.29·5-s − 0.141·7-s − 2.20·9-s + 0.287·11-s + 2.05·15-s − 2.71·17-s − 7.96·19-s + 0.126·21-s − 4.74·23-s + 0.275·25-s + 4.64·27-s + 1.65·29-s − 4.01·31-s − 0.257·33-s + 0.325·35-s − 1.62·37-s + 3.90·41-s + 5.08·43-s + 5.05·45-s − 4.97·47-s − 6.97·49-s + 2.41·51-s − 8.08·53-s − 0.661·55-s + 7.11·57-s + 8.10·59-s + ⋯ |
| L(s) = 1 | − 0.515·3-s − 1.02·5-s − 0.0536·7-s − 0.734·9-s + 0.0868·11-s + 0.529·15-s − 0.657·17-s − 1.82·19-s + 0.0276·21-s − 0.988·23-s + 0.0551·25-s + 0.893·27-s + 0.308·29-s − 0.720·31-s − 0.0447·33-s + 0.0550·35-s − 0.267·37-s + 0.610·41-s + 0.775·43-s + 0.754·45-s − 0.725·47-s − 0.997·49-s + 0.338·51-s − 1.11·53-s − 0.0891·55-s + 0.942·57-s + 1.05·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3632018302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3632018302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.892T + 3T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 + 0.141T + 7T^{2} \) |
| 11 | \( 1 - 0.287T + 11T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 19 | \( 1 + 7.96T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 + 1.62T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 5.08T + 43T^{2} \) |
| 47 | \( 1 + 4.97T + 47T^{2} \) |
| 53 | \( 1 + 8.08T + 53T^{2} \) |
| 59 | \( 1 - 8.10T + 59T^{2} \) |
| 61 | \( 1 - 7.78T + 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 8.82T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 8.17T + 97T^{2} \) |
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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
−8.282191805006717351961136400444, −7.50346440554282814520798167966, −6.60214626712984519773052965376, −6.15023952495865735512481621195, −5.28164049229129898107356623342, −4.35747025196864892242676702868, −3.91721376063971400222093917680, −2.86052423566590440201090672212, −1.89944135227106728484723476074, −0.31501239345256620594832615604,
0.31501239345256620594832615604, 1.89944135227106728484723476074, 2.86052423566590440201090672212, 3.91721376063971400222093917680, 4.35747025196864892242676702868, 5.28164049229129898107356623342, 6.15023952495865735512481621195, 6.60214626712984519773052965376, 7.50346440554282814520798167966, 8.282191805006717351961136400444
