L(s) = 1 | − 2-s + 3.36·3-s + 4-s − 2.80·5-s − 3.36·6-s + 1.62·7-s − 8-s + 8.31·9-s + 2.80·10-s − 4.88·11-s + 3.36·12-s + 3.62·13-s − 1.62·14-s − 9.42·15-s + 16-s + 7.41·17-s − 8.31·18-s + 5.70·19-s − 2.80·20-s + 5.47·21-s + 4.88·22-s + 1.28·23-s − 3.36·24-s + 2.85·25-s − 3.62·26-s + 17.8·27-s + 1.62·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.94·3-s + 0.5·4-s − 1.25·5-s − 1.37·6-s + 0.614·7-s − 0.353·8-s + 2.77·9-s + 0.886·10-s − 1.47·11-s + 0.970·12-s + 1.00·13-s − 0.434·14-s − 2.43·15-s + 0.250·16-s + 1.79·17-s − 1.95·18-s + 1.30·19-s − 0.626·20-s + 1.19·21-s + 1.04·22-s + 0.267·23-s − 0.686·24-s + 0.571·25-s − 0.710·26-s + 3.43·27-s + 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.734397029\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.734397029\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 223 | \( 1 - T \) |
good | 3 | \( 1 - 3.36T + 3T^{2} \) |
| 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 - 1.62T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 - 3.62T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 1.28T + 23T^{2} \) |
| 29 | \( 1 + 4.72T + 29T^{2} \) |
| 31 | \( 1 + 9.52T + 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 + 7.35T + 41T^{2} \) |
| 43 | \( 1 + 5.14T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 - 1.89T + 53T^{2} \) |
| 59 | \( 1 - 2.18T + 59T^{2} \) |
| 61 | \( 1 + 8.49T + 61T^{2} \) |
| 67 | \( 1 + 7.77T + 67T^{2} \) |
| 71 | \( 1 - 1.80T + 71T^{2} \) |
| 73 | \( 1 - 1.22T + 73T^{2} \) |
| 79 | \( 1 + 6.91T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 - 2.46T + 97T^{2} \) |
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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
−10.88059229800180761965375144972, −9.989463869731234297492885703962, −9.089612654065885668525363723403, −8.108153886476696115745684685277, −7.83497070411304641796920203361, −7.30371968017302164016239819326, −5.20487751123882476224416107318, −3.64042356879861026769581718457, −3.12339401458641780686351396388, −1.53905099761482416818232768143,
1.53905099761482416818232768143, 3.12339401458641780686351396388, 3.64042356879861026769581718457, 5.20487751123882476224416107318, 7.30371968017302164016239819326, 7.83497070411304641796920203361, 8.108153886476696115745684685277, 9.089612654065885668525363723403, 9.989463869731234297492885703962, 10.88059229800180761965375144972