| L(s) = 1 | + 3.15·3-s + 1.15·5-s + 6.94·9-s − 2.20·11-s + 1.30·13-s + 3.63·15-s − 17-s + 5.73·19-s + 2.25·23-s − 3.66·25-s + 12.4·27-s − 5.03·29-s − 1.31·31-s − 6.95·33-s + 4.73·37-s + 4.11·39-s − 5.28·41-s + 6.51·43-s + 8.01·45-s + 6.53·47-s − 3.15·51-s + 9.24·53-s − 2.54·55-s + 18.0·57-s + 8.45·59-s + 8.62·61-s + 1.50·65-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 0.515·5-s + 2.31·9-s − 0.665·11-s + 0.361·13-s + 0.939·15-s − 0.242·17-s + 1.31·19-s + 0.469·23-s − 0.733·25-s + 2.39·27-s − 0.935·29-s − 0.236·31-s − 1.21·33-s + 0.778·37-s + 0.658·39-s − 0.826·41-s + 0.994·43-s + 1.19·45-s + 0.952·47-s − 0.441·51-s + 1.26·53-s − 0.343·55-s + 2.39·57-s + 1.10·59-s + 1.10·61-s + 0.186·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.857559752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.857559752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 - 1.15T + 5T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 - 2.25T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 - 6.51T + 43T^{2} \) |
| 47 | \( 1 - 6.53T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 - 8.45T + 59T^{2} \) |
| 61 | \( 1 - 8.62T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 - 2.96T + 71T^{2} \) |
| 73 | \( 1 - 5.95T + 73T^{2} \) |
| 79 | \( 1 + 8.95T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 + 8.74T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.955345401553337994749959185174, −7.50547710270984828526705022644, −6.89747967083913885196101986655, −5.80327983099221630916221610717, −5.13226468356640986607364318903, −4.05694640349757334321519675302, −3.52035973017951256789967659756, −2.63681989475361700956837282972, −2.13093234065339221534366500185, −1.10143898166407722163892776644,
1.10143898166407722163892776644, 2.13093234065339221534366500185, 2.63681989475361700956837282972, 3.52035973017951256789967659756, 4.05694640349757334321519675302, 5.13226468356640986607364318903, 5.80327983099221630916221610717, 6.89747967083913885196101986655, 7.50547710270984828526705022644, 7.955345401553337994749959185174
