| L(s) = 1 | − 2.56·2-s + 0.961·3-s + 4.60·4-s − 2.47·6-s + 2.40·7-s − 6.69·8-s − 2.07·9-s + 2.76·11-s + 4.42·12-s + 1.28·13-s − 6.19·14-s + 7.98·16-s + 6.01·17-s + 5.33·18-s + 2.42·19-s + 2.31·21-s − 7.09·22-s + 4.77·23-s − 6.43·24-s − 3.29·26-s − 4.88·27-s + 11.0·28-s + 7.77·29-s − 4.82·31-s − 7.14·32-s + 2.65·33-s − 15.4·34-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.555·3-s + 2.30·4-s − 1.00·6-s + 0.910·7-s − 2.36·8-s − 0.691·9-s + 0.832·11-s + 1.27·12-s + 0.355·13-s − 1.65·14-s + 1.99·16-s + 1.45·17-s + 1.25·18-s + 0.556·19-s + 0.505·21-s − 1.51·22-s + 0.996·23-s − 1.31·24-s − 0.646·26-s − 0.939·27-s + 2.09·28-s + 1.44·29-s − 0.865·31-s − 1.26·32-s + 0.462·33-s − 2.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.412625859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412625859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 - 0.961T + 3T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 - 7.77T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 - 9.25T + 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 - 7.89T + 43T^{2} \) |
| 47 | \( 1 - 4.18T + 47T^{2} \) |
| 53 | \( 1 - 2.01T + 53T^{2} \) |
| 59 | \( 1 - 7.91T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 - 1.72T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 8.98T + 73T^{2} \) |
| 79 | \( 1 + 3.90T + 79T^{2} \) |
| 83 | \( 1 + 3.72T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 4.15T + 97T^{2} \) |
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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
−8.305907861830373334855052910103, −7.56789406054261094191371027986, −7.17763031164630789631070798557, −6.10883689095435240784082400231, −5.54934005386631685796753694359, −4.31801774728430291067643375811, −3.17216320324330199548673144594, −2.57978699588815724053383625348, −1.43741592064288087051125964894, −0.908938679598441019402124722086,
0.908938679598441019402124722086, 1.43741592064288087051125964894, 2.57978699588815724053383625348, 3.17216320324330199548673144594, 4.31801774728430291067643375811, 5.54934005386631685796753694359, 6.10883689095435240784082400231, 7.17763031164630789631070798557, 7.56789406054261094191371027986, 8.305907861830373334855052910103
