L(s) = 1 | + 1.16·3-s + 3.89·5-s − 0.552·7-s − 1.64·9-s + 2.70·11-s − 0.196·13-s + 4.52·15-s + 17-s − 0.578·19-s − 0.643·21-s + 6.00·23-s + 10.1·25-s − 5.40·27-s − 0.846·29-s + 1.20·31-s + 3.15·33-s − 2.15·35-s + 10.3·37-s − 0.228·39-s + 0.830·41-s + 7.72·43-s − 6.39·45-s + 3.03·47-s − 6.69·49-s + 1.16·51-s − 11.1·53-s + 10.5·55-s + ⋯ |
L(s) = 1 | + 0.672·3-s + 1.73·5-s − 0.208·7-s − 0.548·9-s + 0.816·11-s − 0.0544·13-s + 1.16·15-s + 0.242·17-s − 0.132·19-s − 0.140·21-s + 1.25·23-s + 2.02·25-s − 1.04·27-s − 0.157·29-s + 0.217·31-s + 0.548·33-s − 0.363·35-s + 1.69·37-s − 0.0365·39-s + 0.129·41-s + 1.17·43-s − 0.953·45-s + 0.442·47-s − 0.956·49-s + 0.162·51-s − 1.53·53-s + 1.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.970534326\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.970534326\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 1.16T + 3T^{2} \) |
| 5 | \( 1 - 3.89T + 5T^{2} \) |
| 7 | \( 1 + 0.552T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 0.196T + 13T^{2} \) |
| 19 | \( 1 + 0.578T + 19T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 + 0.846T + 29T^{2} \) |
| 31 | \( 1 - 1.20T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 0.830T + 41T^{2} \) |
| 43 | \( 1 - 7.72T + 43T^{2} \) |
| 47 | \( 1 - 3.03T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 61 | \( 1 - 4.75T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 2.17T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 + 6.27T + 83T^{2} \) |
| 89 | \( 1 + 0.770T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−7.87443843107038736504311034848, −7.09068990110152053977392502620, −6.23729626327786296026529774667, −5.95710389192256984817911869038, −5.14115780819237493222269389669, −4.30377320287862622605755288556, −3.19792983486557584861595921217, −2.68757760140687496573286232263, −1.88897281336000433480435564546, −1.00021846178588236950545498424,
1.00021846178588236950545498424, 1.88897281336000433480435564546, 2.68757760140687496573286232263, 3.19792983486557584861595921217, 4.30377320287862622605755288556, 5.14115780819237493222269389669, 5.95710389192256984817911869038, 6.23729626327786296026529774667, 7.09068990110152053977392502620, 7.87443843107038736504311034848