| L(s) = 1 | − 1.53·3-s − 0.551·5-s − 1.16·7-s − 0.629·9-s + 4.36·11-s + 4.01·13-s + 0.848·15-s − 4.62·17-s + 6.90·19-s + 1.78·21-s + 9.20·23-s − 4.69·25-s + 5.58·27-s − 8.36·29-s − 7.71·31-s − 6.71·33-s + 0.640·35-s + 3.55·37-s − 6.18·39-s + 7.65·41-s − 10.3·43-s + 0.347·45-s − 5.43·47-s − 5.65·49-s + 7.12·51-s − 4.47·53-s − 2.40·55-s + ⋯ |
| L(s) = 1 | − 0.888·3-s − 0.246·5-s − 0.438·7-s − 0.209·9-s + 1.31·11-s + 1.11·13-s + 0.219·15-s − 1.12·17-s + 1.58·19-s + 0.390·21-s + 1.91·23-s − 0.939·25-s + 1.07·27-s − 1.55·29-s − 1.38·31-s − 1.16·33-s + 0.108·35-s + 0.583·37-s − 0.990·39-s + 1.19·41-s − 1.58·43-s + 0.0517·45-s − 0.792·47-s − 0.807·49-s + 0.997·51-s − 0.615·53-s − 0.324·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.187335520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187335520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 + 0.551T + 5T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 23 | \( 1 - 9.20T + 23T^{2} \) |
| 29 | \( 1 + 8.36T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 - 3.55T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 - 5.04T + 61T^{2} \) |
| 67 | \( 1 + 7.56T + 67T^{2} \) |
| 71 | \( 1 - 3.20T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 5.61T + 83T^{2} \) |
| 89 | \( 1 - 6.23T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−8.587212388207051562289932665060, −7.54481968780138032503499446650, −6.80627230508880038969921556100, −6.26337335756227028938244073162, −5.55470107778637363894055680822, −4.81206851035278400693931271118, −3.72720949563836997368501883985, −3.25499789750445041533888590592, −1.72280432306875941171124850064, −0.67300804127092909355412776562,
0.67300804127092909355412776562, 1.72280432306875941171124850064, 3.25499789750445041533888590592, 3.72720949563836997368501883985, 4.81206851035278400693931271118, 5.55470107778637363894055680822, 6.26337335756227028938244073162, 6.80627230508880038969921556100, 7.54481968780138032503499446650, 8.587212388207051562289932665060
