| L(s) = 1 | − 1.86·3-s − 2.88·5-s + 3.98·7-s + 0.493·9-s − 11-s + 5.34·13-s + 5.38·15-s + 1.15·17-s − 6.56·19-s − 7.44·21-s − 9.46·23-s + 3.31·25-s + 4.68·27-s + 9.10·29-s + 3.07·31-s + 1.86·33-s − 11.4·35-s − 2.28·37-s − 9.99·39-s − 9.82·41-s − 6.85·43-s − 1.42·45-s + 5.58·47-s + 8.87·49-s − 2.15·51-s − 3.94·53-s + 2.88·55-s + ⋯ |
| L(s) = 1 | − 1.07·3-s − 1.28·5-s + 1.50·7-s + 0.164·9-s − 0.301·11-s + 1.48·13-s + 1.39·15-s + 0.279·17-s − 1.50·19-s − 1.62·21-s − 1.97·23-s + 0.662·25-s + 0.901·27-s + 1.69·29-s + 0.552·31-s + 0.325·33-s − 1.94·35-s − 0.375·37-s − 1.60·39-s − 1.53·41-s − 1.04·43-s − 0.212·45-s + 0.815·47-s + 1.26·49-s − 0.301·51-s − 0.541·53-s + 0.388·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 1.86T + 3T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 - 3.98T + 7T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 + 9.46T + 23T^{2} \) |
| 29 | \( 1 - 9.10T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 + 2.28T + 37T^{2} \) |
| 41 | \( 1 + 9.82T + 41T^{2} \) |
| 43 | \( 1 + 6.85T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 4.49T + 73T^{2} \) |
| 79 | \( 1 - 2.62T + 79T^{2} \) |
| 83 | \( 1 - 2.65T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−8.016682296142336135067957327686, −6.89046756383675006497205034413, −6.28930174950832905990187725573, −5.55531062400351761634435499180, −4.74830199783361399232928582147, −4.24218696882958589666454683975, −3.50974828607971936963659286186, −2.13808143318920742441196605294, −1.08734708593797924556514279269, 0,
1.08734708593797924556514279269, 2.13808143318920742441196605294, 3.50974828607971936963659286186, 4.24218696882958589666454683975, 4.74830199783361399232928582147, 5.55531062400351761634435499180, 6.28930174950832905990187725573, 6.89046756383675006497205034413, 8.016682296142336135067957327686
