| L(s) = 1 | − 0.492·3-s + 0.923·5-s − 1.84·7-s − 2.75·9-s − 11-s − 6.93·13-s − 0.454·15-s − 2.04·17-s − 3.74·19-s + 0.907·21-s − 9.37·23-s − 4.14·25-s + 2.83·27-s − 9.05·29-s + 2.98·31-s + 0.492·33-s − 1.70·35-s + 4.12·37-s + 3.41·39-s − 1.47·41-s − 4.51·43-s − 2.54·45-s + 8.14·47-s − 3.60·49-s + 1.00·51-s + 53-s − 0.923·55-s + ⋯ |
| L(s) = 1 | − 0.284·3-s + 0.413·5-s − 0.696·7-s − 0.919·9-s − 0.301·11-s − 1.92·13-s − 0.117·15-s − 0.496·17-s − 0.858·19-s + 0.198·21-s − 1.95·23-s − 0.829·25-s + 0.545·27-s − 1.68·29-s + 0.535·31-s + 0.0857·33-s − 0.287·35-s + 0.678·37-s + 0.546·39-s − 0.230·41-s − 0.688·43-s − 0.379·45-s + 1.18·47-s − 0.514·49-s + 0.141·51-s + 0.137·53-s − 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1939287212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1939287212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 0.492T + 3T^{2} \) |
| 5 | \( 1 - 0.923T + 5T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 13 | \( 1 + 6.93T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 9.37T + 23T^{2} \) |
| 29 | \( 1 + 9.05T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 + 4.51T + 43T^{2} \) |
| 47 | \( 1 - 8.14T + 47T^{2} \) |
| 59 | \( 1 - 2.71T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 + 4.40T + 71T^{2} \) |
| 73 | \( 1 - 0.530T + 73T^{2} \) |
| 79 | \( 1 - 7.81T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 0.0829T + 89T^{2} \) |
| 97 | \( 1 + 8.81T + 97T^{2} \) |
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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
−7.71995376352193579467288177216, −6.98835807384807219324211941981, −6.13707833900929647602668312714, −5.84204888128717125651342800836, −5.02147834055991638286062114149, −4.28055355659035386178396652887, −3.40758818499448322147552162957, −2.34716872205512039791756698660, −2.12106057582462024864875050125, −0.19347096697505895012888086693,
0.19347096697505895012888086693, 2.12106057582462024864875050125, 2.34716872205512039791756698660, 3.40758818499448322147552162957, 4.28055355659035386178396652887, 5.02147834055991638286062114149, 5.84204888128717125651342800836, 6.13707833900929647602668312714, 6.98835807384807219324211941981, 7.71995376352193579467288177216
