| L(s) = 1 | + 2.76·2-s + 5.62·4-s + 1.86·7-s + 10.0·8-s − 11-s + 4.62·13-s + 5.14·14-s + 16.4·16-s − 2.49·17-s − 5.38·19-s − 2.76·22-s − 7.14·23-s + 12.7·26-s + 10.4·28-s + 3.52·29-s + 8.62·31-s + 25.2·32-s − 6.87·34-s − 8.87·37-s − 14.8·38-s + 0.761·41-s − 7.40·43-s − 5.62·44-s − 19.7·46-s + 0.373·47-s − 3.52·49-s + 26.0·52-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 2.81·4-s + 0.704·7-s + 3.54·8-s − 0.301·11-s + 1.28·13-s + 1.37·14-s + 4.10·16-s − 0.604·17-s − 1.23·19-s − 0.588·22-s − 1.49·23-s + 2.50·26-s + 1.98·28-s + 0.654·29-s + 1.54·31-s + 4.46·32-s − 1.17·34-s − 1.45·37-s − 2.41·38-s + 0.118·41-s − 1.12·43-s − 0.848·44-s − 2.91·46-s + 0.0545·47-s − 0.503·49-s + 3.60·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.176312826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.176312826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 13 | \( 1 - 4.62T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 + 8.87T + 37T^{2} \) |
| 41 | \( 1 - 0.761T + 41T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 - 0.373T + 47T^{2} \) |
| 53 | \( 1 - 5.45T + 53T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 6.77T + 73T^{2} \) |
| 79 | \( 1 + 6.01T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 9.04T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.458141850878130387782702596343, −8.150664633216474061125056018075, −6.96025917029681205196735449243, −6.35738895911670387034565210788, −5.75004515543937225691441764321, −4.79706910050281371658088931911, −4.25074388172314136697070592345, −3.46811085292247000938630381903, −2.38985769805978353573635527817, −1.58848550648761436895596697014,
1.58848550648761436895596697014, 2.38985769805978353573635527817, 3.46811085292247000938630381903, 4.25074388172314136697070592345, 4.79706910050281371658088931911, 5.75004515543937225691441764321, 6.35738895911670387034565210788, 6.96025917029681205196735449243, 8.150664633216474061125056018075, 8.458141850878130387782702596343
