| L(s) = 1 | − 1.43·3-s + 5-s + 3.08·7-s − 0.950·9-s + 6.46·11-s − 3.95·13-s − 1.43·15-s − 3.43·17-s − 3.08·19-s − 4.41·21-s − 23-s + 25-s + 5.65·27-s − 0.863·29-s − 5.95·31-s − 9.26·33-s + 3.08·35-s + 7.03·37-s + 5.65·39-s + 5.60·41-s − 8·43-s − 0.950·45-s + 3.90·47-s + 2.53·49-s + 4.91·51-s + 6·53-s + 6.46·55-s + ⋯ |
| L(s) = 1 | − 0.826·3-s + 0.447·5-s + 1.16·7-s − 0.316·9-s + 1.95·11-s − 1.09·13-s − 0.369·15-s − 0.832·17-s − 0.708·19-s − 0.964·21-s − 0.208·23-s + 0.200·25-s + 1.08·27-s − 0.160·29-s − 1.06·31-s − 1.61·33-s + 0.521·35-s + 1.15·37-s + 0.905·39-s + 0.875·41-s − 1.21·43-s − 0.141·45-s + 0.569·47-s + 0.361·49-s + 0.687·51-s + 0.824·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.767830785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.767830785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.43T + 3T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 13 | \( 1 + 3.95T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 29 | \( 1 + 0.863T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6.86T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.933205955193571812985985879215, −6.93153461407351123472291171001, −6.55920212659188382082974673358, −5.78651923104245643403329372777, −5.11294799573139502894573729276, −4.46751694572465156646127432685, −3.80678464993306920981008236594, −2.42302023027486780905590764989, −1.77116050911599780063759440301, −0.71188258339958861384617998866,
0.71188258339958861384617998866, 1.77116050911599780063759440301, 2.42302023027486780905590764989, 3.80678464993306920981008236594, 4.46751694572465156646127432685, 5.11294799573139502894573729276, 5.78651923104245643403329372777, 6.55920212659188382082974673358, 6.93153461407351123472291171001, 7.933205955193571812985985879215
