| L(s) = 1 | − 1.30·3-s + 4.26·5-s − 3.65·7-s − 1.29·9-s − 3.47·11-s + 2.63·13-s − 5.56·15-s + 6.61·17-s − 19-s + 4.76·21-s + 6.79·23-s + 13.1·25-s + 5.60·27-s − 0.634·29-s − 6.95·31-s + 4.53·33-s − 15.5·35-s + 2.05·37-s − 3.44·39-s − 5.58·41-s + 0.274·43-s − 5.53·45-s + 6.20·47-s + 6.34·49-s − 8.63·51-s − 6.21·53-s − 14.8·55-s + ⋯ |
| L(s) = 1 | − 0.753·3-s + 1.90·5-s − 1.38·7-s − 0.432·9-s − 1.04·11-s + 0.731·13-s − 1.43·15-s + 1.60·17-s − 0.229·19-s + 1.03·21-s + 1.41·23-s + 2.63·25-s + 1.07·27-s − 0.117·29-s − 1.24·31-s + 0.788·33-s − 2.63·35-s + 0.337·37-s − 0.550·39-s − 0.872·41-s + 0.0418·43-s − 0.825·45-s + 0.904·47-s + 0.905·49-s − 1.20·51-s − 0.853·53-s − 1.99·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.666838789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666838789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 23 | \( 1 - 6.79T + 23T^{2} \) |
| 29 | \( 1 + 0.634T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 - 2.05T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 - 0.274T + 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 + 6.21T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 5.83T + 73T^{2} \) |
| 83 | \( 1 - 0.637T + 83T^{2} \) |
| 89 | \( 1 + 4.90T + 89T^{2} \) |
| 97 | \( 1 - 9.14T + 97T^{2} \) |
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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
−8.108205367011037077338273889411, −7.06836065788218563757863781145, −6.44297986041302372133968031212, −5.76574391638105726322136518638, −5.59436139828908729569831637074, −4.83968020724473230309408313891, −3.18018945429045129732601514030, −3.00490305996912393040905671182, −1.79119618438719571525223003075, −0.70332519031843849074108543221,
0.70332519031843849074108543221, 1.79119618438719571525223003075, 3.00490305996912393040905671182, 3.18018945429045129732601514030, 4.83968020724473230309408313891, 5.59436139828908729569831637074, 5.76574391638105726322136518638, 6.44297986041302372133968031212, 7.06836065788218563757863781145, 8.108205367011037077338273889411
