| L(s) = 1 | + 1.27·5-s + i·7-s − 6.57i·11-s + 4.05i·13-s − 6.09i·17-s + 2.08·19-s − 6.85·23-s − 3.37·25-s − 6.53·29-s + 3.26i·31-s + 1.27i·35-s + 2.95i·37-s + 3.35i·41-s − 10.9·43-s − 7.12·47-s + ⋯ |
| L(s) = 1 | + 0.570·5-s + 0.377i·7-s − 1.98i·11-s + 1.12i·13-s − 1.47i·17-s + 0.477·19-s − 1.42·23-s − 0.674·25-s − 1.21·29-s + 0.587i·31-s + 0.215i·35-s + 0.485i·37-s + 0.523i·41-s − 1.67·43-s − 1.03·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2340378758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2340378758\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| good | 5 | \( 1 - 1.27T + 5T^{2} \) |
| 11 | \( 1 + 6.57iT - 11T^{2} \) |
| 13 | \( 1 - 4.05iT - 13T^{2} \) |
| 17 | \( 1 + 6.09iT - 17T^{2} \) |
| 19 | \( 1 - 2.08T + 19T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 + 6.53T + 29T^{2} \) |
| 31 | \( 1 - 3.26iT - 31T^{2} \) |
| 37 | \( 1 - 2.95iT - 37T^{2} \) |
| 41 | \( 1 - 3.35iT - 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 7.75iT - 59T^{2} \) |
| 61 | \( 1 - 12.0iT - 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 + 0.849iT - 79T^{2} \) |
| 83 | \( 1 - 15.8iT - 83T^{2} \) |
| 89 | \( 1 + 4.06iT - 89T^{2} \) |
| 97 | \( 1 + 4.90T + 97T^{2} \) |
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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
−8.474670654957056051338662944326, −7.72626883648052630579073553754, −6.84915715676254126630693127792, −6.14769322346767331279996349617, −5.62177974883195678078151498958, −4.93439595518967938900356390395, −3.85811643988854994364595112616, −3.14630783895056738216945727673, −2.27048124687890943092640491731, −1.28999644412899235549868757988,
0.05435972671418144870451048312, 1.82784888773182119498559785929, 1.96407207471704701164791342227, 3.43341463055469813888061937688, 4.05498794056853836389176744668, 4.93056333575656050571848224648, 5.66129414188164191424229524850, 6.30479723682129267892157052861, 7.13016519853756850565238782054, 7.83755869476044071818695762145
