| L(s) = 1 | − 2.43·2-s + 3-s + 3.95·4-s − 2.28·5-s − 2.43·6-s − 3.60·7-s − 4.75·8-s + 9-s + 5.56·10-s − 1.92·11-s + 3.95·12-s + 6.25·13-s + 8.79·14-s − 2.28·15-s + 3.70·16-s + 0.877·17-s − 2.43·18-s + 4.73·19-s − 9.00·20-s − 3.60·21-s + 4.69·22-s + 9.12·23-s − 4.75·24-s + 0.199·25-s − 15.2·26-s + 27-s − 14.2·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 0.577·3-s + 1.97·4-s − 1.01·5-s − 0.995·6-s − 1.36·7-s − 1.68·8-s + 0.333·9-s + 1.75·10-s − 0.580·11-s + 1.14·12-s + 1.73·13-s + 2.35·14-s − 0.588·15-s + 0.926·16-s + 0.212·17-s − 0.574·18-s + 1.08·19-s − 2.01·20-s − 0.787·21-s + 1.00·22-s + 1.90·23-s − 0.971·24-s + 0.0398·25-s − 2.99·26-s + 0.192·27-s − 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5520128366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5520128366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 - 6.25T + 13T^{2} \) |
| 17 | \( 1 - 0.877T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 - 9.12T + 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 + 0.598T + 31T^{2} \) |
| 37 | \( 1 - 5.44T + 37T^{2} \) |
| 41 | \( 1 + 3.45T + 41T^{2} \) |
| 43 | \( 1 - 6.49T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 + 0.168T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 0.264T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 8.95T + 79T^{2} \) |
| 83 | \( 1 + 6.01T + 83T^{2} \) |
| 89 | \( 1 + 2.09T + 89T^{2} \) |
| 97 | \( 1 + 2.80T + 97T^{2} \) |
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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
−10.93934582200481675324370741035, −10.12378538806441854162366395530, −9.124676195573470501420807517051, −8.732661927436860057892850205377, −7.59207313658424699229821574505, −7.15878305927626800688114859156, −5.91155758376351747784608429621, −3.75639292049642095315199411348, −2.86568615436856508692502433221, −0.892571915075470521168122346749,
0.892571915075470521168122346749, 2.86568615436856508692502433221, 3.75639292049642095315199411348, 5.91155758376351747784608429621, 7.15878305927626800688114859156, 7.59207313658424699229821574505, 8.732661927436860057892850205377, 9.124676195573470501420807517051, 10.12378538806441854162366395530, 10.93934582200481675324370741035
