| L(s) = 1 | − 2.87·3-s − 2.95·5-s − 5.20·7-s + 5.28·9-s + 6.14·11-s − 2.39·13-s + 8.51·15-s − 6.97·17-s − 3.26·19-s + 14.9·21-s − 2.38·23-s + 3.75·25-s − 6.58·27-s − 7.00·29-s − 0.758·31-s − 17.6·33-s + 15.4·35-s − 2.09·37-s + 6.90·39-s + 11.4·41-s + 4.49·43-s − 15.6·45-s − 3.55·47-s + 20.0·49-s + 20.0·51-s + 5.35·53-s − 18.1·55-s + ⋯ |
| L(s) = 1 | − 1.66·3-s − 1.32·5-s − 1.96·7-s + 1.76·9-s + 1.85·11-s − 0.665·13-s + 2.19·15-s − 1.69·17-s − 0.747·19-s + 3.26·21-s − 0.497·23-s + 0.751·25-s − 1.26·27-s − 1.30·29-s − 0.136·31-s − 3.07·33-s + 2.60·35-s − 0.345·37-s + 1.10·39-s + 1.78·41-s + 0.685·43-s − 2.33·45-s − 0.517·47-s + 2.86·49-s + 2.81·51-s + 0.735·53-s − 2.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 2.87T + 3T^{2} \) |
| 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 + 5.20T + 7T^{2} \) |
| 11 | \( 1 - 6.14T + 11T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 + 6.97T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 2.38T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 + 0.758T + 31T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4.49T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 - 5.35T + 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 - 0.141T + 61T^{2} \) |
| 67 | \( 1 - 9.78T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 6.27T + 73T^{2} \) |
| 79 | \( 1 + 9.72T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 - 1.53T + 97T^{2} \) |
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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
−7.11097909168827532415365199835, −7.05285155106307962546779713404, −6.27505530565193548208148308207, −5.92114152978981333100533872026, −4.69902746282224287740943036235, −3.91481023116075160391318346605, −3.79460933552955267116258942770, −2.28164323032773743757127071399, −0.68298164132013736233165643503, 0,
0.68298164132013736233165643503, 2.28164323032773743757127071399, 3.79460933552955267116258942770, 3.91481023116075160391318346605, 4.69902746282224287740943036235, 5.92114152978981333100533872026, 6.27505530565193548208148308207, 7.05285155106307962546779713404, 7.11097909168827532415365199835
