| L(s) = 1 | − 2.74·3-s + 5-s − 7-s + 4.50·9-s − 3.92·11-s + 13-s − 2.74·15-s − 3.43·17-s − 1.39·19-s + 2.74·21-s − 4.90·23-s + 25-s − 4.13·27-s + 4.16·29-s − 1.75·31-s + 10.7·33-s − 35-s + 5.10·37-s − 2.74·39-s − 0.528·41-s − 4.03·43-s + 4.50·45-s + 5.27·47-s + 49-s + 9.41·51-s − 10.6·53-s − 3.92·55-s + ⋯ |
| L(s) = 1 | − 1.58·3-s + 0.447·5-s − 0.377·7-s + 1.50·9-s − 1.18·11-s + 0.277·13-s − 0.707·15-s − 0.833·17-s − 0.320·19-s + 0.597·21-s − 1.02·23-s + 0.200·25-s − 0.796·27-s + 0.773·29-s − 0.316·31-s + 1.87·33-s − 0.169·35-s + 0.840·37-s − 0.438·39-s − 0.0825·41-s − 0.615·43-s + 0.672·45-s + 0.769·47-s + 0.142·49-s + 1.31·51-s − 1.46·53-s − 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6166943462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6166943462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.74T + 3T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 + 0.528T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 - 5.27T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 8.01T + 59T^{2} \) |
| 61 | \( 1 + 4.75T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 3.05T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 3.50T + 89T^{2} \) |
| 97 | \( 1 - 4.90T + 97T^{2} \) |
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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
−8.512696153573428771218334185464, −7.66992280109476526547993210976, −6.78141996208718407934231281460, −6.17608028871920141436552364513, −5.68274841056027616669042018142, −4.88058867364492546197949717347, −4.23320418905989631972122117201, −2.91509024097288717075643370924, −1.85484302850842171838853960377, −0.48333472831640626318437016445,
0.48333472831640626318437016445, 1.85484302850842171838853960377, 2.91509024097288717075643370924, 4.23320418905989631972122117201, 4.88058867364492546197949717347, 5.68274841056027616669042018142, 6.17608028871920141436552364513, 6.78141996208718407934231281460, 7.66992280109476526547993210976, 8.512696153573428771218334185464
