| L(s) = 1 | + 3-s + 1.84·5-s + 2.96·7-s + 9-s + 3.06·11-s + 0.260·13-s + 1.84·15-s + 4.67·17-s + 1.84·19-s + 2.96·21-s − 6.65·23-s − 1.59·25-s + 27-s − 6.07·29-s + 3.09·31-s + 3.06·33-s + 5.46·35-s − 6.26·37-s + 0.260·39-s − 2.16·41-s + 7.03·43-s + 1.84·45-s + 7.08·47-s + 1.77·49-s + 4.67·51-s + 4.74·53-s + 5.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.825·5-s + 1.11·7-s + 0.333·9-s + 0.922·11-s + 0.0723·13-s + 0.476·15-s + 1.13·17-s + 0.423·19-s + 0.646·21-s − 1.38·23-s − 0.318·25-s + 0.192·27-s − 1.12·29-s + 0.556·31-s + 0.532·33-s + 0.924·35-s − 1.03·37-s + 0.0417·39-s − 0.338·41-s + 1.07·43-s + 0.275·45-s + 1.03·47-s + 0.254·49-s + 0.654·51-s + 0.652·53-s + 0.761·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.120081345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.120081345\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 - 0.260T + 13T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 + 6.65T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 6.26T + 37T^{2} \) |
| 41 | \( 1 + 2.16T + 41T^{2} \) |
| 43 | \( 1 - 7.03T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 - 4.74T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 4.09T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 7.96T + 73T^{2} \) |
| 79 | \( 1 + 4.67T + 79T^{2} \) |
| 83 | \( 1 + 5.56T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 1.21T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.976347858117353468451645378777, −7.24430518709099761828346749880, −6.46561022795824995739076776577, −5.58026721016236792487329556575, −5.25198106715000441858039914954, −4.04397650234032632749436627409, −3.70967254497796209230053582564, −2.44261544789218967309838068815, −1.81763412715701809334937515256, −1.07167102361148881915169354920,
1.07167102361148881915169354920, 1.81763412715701809334937515256, 2.44261544789218967309838068815, 3.70967254497796209230053582564, 4.04397650234032632749436627409, 5.25198106715000441858039914954, 5.58026721016236792487329556575, 6.46561022795824995739076776577, 7.24430518709099761828346749880, 7.976347858117353468451645378777
