| L(s) = 1 | − 3.07·3-s − 5-s − 2.07·7-s + 6.48·9-s − 5.07·11-s + 3.48·13-s + 3.07·15-s + 6.48·17-s + 7.48·19-s + 6.40·21-s + 23-s + 25-s − 10.7·27-s + 1.56·29-s + 0.0791·31-s + 15.6·33-s + 2.07·35-s + 9.71·37-s − 10.7·39-s − 0.480·41-s − 8·43-s − 6.48·45-s + 6.96·47-s − 2.67·49-s − 19.9·51-s − 11.7·53-s + 5.07·55-s + ⋯ |
| L(s) = 1 | − 1.77·3-s − 0.447·5-s − 0.785·7-s + 2.16·9-s − 1.53·11-s + 0.965·13-s + 0.795·15-s + 1.57·17-s + 1.71·19-s + 1.39·21-s + 0.208·23-s + 0.200·25-s − 2.06·27-s + 0.289·29-s + 0.0142·31-s + 2.72·33-s + 0.351·35-s + 1.59·37-s − 1.71·39-s − 0.0751·41-s − 1.21·43-s − 0.966·45-s + 1.01·47-s − 0.382·49-s − 2.79·51-s − 1.60·53-s + 0.684·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7666653994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7666653994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 + 5.07T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 19 | \( 1 - 7.48T + 19T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 0.0791T + 31T^{2} \) |
| 37 | \( 1 - 9.71T + 37T^{2} \) |
| 41 | \( 1 + 0.480T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 7.88T + 61T^{2} \) |
| 67 | \( 1 - 9.71T + 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 4.59T + 83T^{2} \) |
| 89 | \( 1 - 8.31T + 89T^{2} \) |
| 97 | \( 1 - 7.23T + 97T^{2} \) |
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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
−7.58362781486109981952708316017, −7.27797525486608372860102688235, −6.24422591660711639195267012790, −5.82861657140085153487040022572, −5.22868676482276634584558203953, −4.63168721333929072198155220434, −3.52189124998402295078262230321, −2.95554500334128519350724674562, −1.29190332526380956622966325380, −0.55120327721720638571237250741,
0.55120327721720638571237250741, 1.29190332526380956622966325380, 2.95554500334128519350724674562, 3.52189124998402295078262230321, 4.63168721333929072198155220434, 5.22868676482276634584558203953, 5.82861657140085153487040022572, 6.24422591660711639195267012790, 7.27797525486608372860102688235, 7.58362781486109981952708316017
