| L(s) = 1 | + 2.23·5-s + 0.540·11-s − 2.62·13-s − 1.47·17-s + 4.03·19-s − 3.16·23-s − 0.540·29-s − 2.62·31-s + 4.70·37-s + 2.23·41-s + 8.70·43-s + 4.52·47-s + 11.6·53-s + 1.20·55-s + 11.9·59-s + 9.69·61-s − 5.86·65-s + 3.70·67-s − 13.7·71-s − 1.41·73-s + 5·79-s + 3.76·83-s − 3.29·85-s + 6.76·89-s + 9.02·95-s − 9.69·97-s + 9.70·101-s + ⋯ |
| L(s) = 1 | + 0.999·5-s + 0.162·11-s − 0.727·13-s − 0.357·17-s + 0.925·19-s − 0.659·23-s − 0.100·29-s − 0.470·31-s + 0.774·37-s + 0.349·41-s + 1.32·43-s + 0.660·47-s + 1.59·53-s + 0.162·55-s + 1.55·59-s + 1.24·61-s − 0.727·65-s + 0.453·67-s − 1.62·71-s − 0.165·73-s + 0.562·79-s + 0.413·83-s − 0.357·85-s + 0.716·89-s + 0.925·95-s − 0.984·97-s + 0.966·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.346844624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346844624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2.23T + 5T^{2} \) |
| 11 | \( 1 - 0.540T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 0.540T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 - 4.52T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 9.69T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 + 9.69T + 97T^{2} \) |
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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
−8.180238668033566688178467323278, −7.37847963732750566054944422133, −6.82905942718138660507718467728, −5.77167268866410672461825963350, −5.58361912100299143318788785465, −4.53300535380620074408842497768, −3.75142096399955907881396731578, −2.59505378192517201629801839598, −2.04391750517139598042381742974, −0.830157153368997862665909012101,
0.830157153368997862665909012101, 2.04391750517139598042381742974, 2.59505378192517201629801839598, 3.75142096399955907881396731578, 4.53300535380620074408842497768, 5.58361912100299143318788785465, 5.77167268866410672461825963350, 6.82905942718138660507718467728, 7.37847963732750566054944422133, 8.180238668033566688178467323278
