| L(s) = 1 | − 3-s + 3.61·5-s − 7-s + 9-s − 11-s + 0.618·13-s − 3.61·15-s − 5.47·17-s − 4.23·19-s + 21-s − 23-s + 8.09·25-s − 27-s + 1.76·29-s + 8.70·31-s + 33-s − 3.61·35-s + 0.236·37-s − 0.618·39-s − 3.47·41-s + 3.85·43-s + 3.61·45-s − 11.7·47-s + 49-s + 5.47·51-s − 0.0901·53-s − 3.61·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.61·5-s − 0.377·7-s + 0.333·9-s − 0.301·11-s + 0.171·13-s − 0.934·15-s − 1.32·17-s − 0.971·19-s + 0.218·21-s − 0.208·23-s + 1.61·25-s − 0.192·27-s + 0.327·29-s + 1.56·31-s + 0.174·33-s − 0.611·35-s + 0.0388·37-s − 0.0989·39-s − 0.542·41-s + 0.587·43-s + 0.539·45-s − 1.70·47-s + 0.142·49-s + 0.766·51-s − 0.0123·53-s − 0.487·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.929375306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.929375306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 0.618T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.0901T + 53T^{2} \) |
| 59 | \( 1 - 3.61T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 - 8.09T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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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
−7.904630603513350172279047108233, −6.71782593606122437679922198866, −6.42017326403305414967239293656, −6.01263300083463931782758747007, −5.00353327773323228069621979639, −4.64778310095972960498113334647, −3.48209687558609221778311485796, −2.38248205655344413828348849842, −1.94920769400065955433230506405, −0.69882725097719064123446083580,
0.69882725097719064123446083580, 1.94920769400065955433230506405, 2.38248205655344413828348849842, 3.48209687558609221778311485796, 4.64778310095972960498113334647, 5.00353327773323228069621979639, 6.01263300083463931782758747007, 6.42017326403305414967239293656, 6.71782593606122437679922198866, 7.904630603513350172279047108233
