| L(s) = 1 | + 0.712·2-s + 3-s − 1.49·4-s − 0.415·5-s + 0.712·6-s − 2.48·8-s + 9-s − 0.295·10-s + 2.75·11-s − 1.49·12-s − 4.20·13-s − 0.415·15-s + 1.21·16-s + 4.39·17-s + 0.712·18-s − 1.26·19-s + 0.619·20-s + 1.96·22-s + 4.53·23-s − 2.48·24-s − 4.82·25-s − 2.99·26-s + 27-s − 5.42·29-s − 0.295·30-s − 8.86·31-s + 5.84·32-s + ⋯ |
| L(s) = 1 | + 0.503·2-s + 0.577·3-s − 0.746·4-s − 0.185·5-s + 0.290·6-s − 0.879·8-s + 0.333·9-s − 0.0935·10-s + 0.831·11-s − 0.430·12-s − 1.16·13-s − 0.107·15-s + 0.302·16-s + 1.06·17-s + 0.167·18-s − 0.290·19-s + 0.138·20-s + 0.419·22-s + 0.944·23-s − 0.508·24-s − 0.965·25-s − 0.587·26-s + 0.192·27-s − 1.00·29-s − 0.0540·30-s − 1.59·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.712T + 2T^{2} \) |
| 5 | \( 1 + 0.415T + 5T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 + 4.20T + 13T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 + 5.42T + 29T^{2} \) |
| 31 | \( 1 + 8.86T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 43 | \( 1 - 4.38T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2.30T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 1.10T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 + 3.91T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 4.04T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 - 7.88T + 97T^{2} \) |
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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
−7.45209810765321284468848273935, −7.38896284746862864238025183657, −6.11432026263251391960239813546, −5.47331087600216845258013620758, −4.74262338638421627072978755630, −3.93006114114935872748011592794, −3.47550609597114637571955442911, −2.52945263807440330663330456446, −1.39355978312033456804145312181, 0,
1.39355978312033456804145312181, 2.52945263807440330663330456446, 3.47550609597114637571955442911, 3.93006114114935872748011592794, 4.74262338638421627072978755630, 5.47331087600216845258013620758, 6.11432026263251391960239813546, 7.38896284746862864238025183657, 7.45209810765321284468848273935
