L(s) = 1 | − 1.92·3-s − 1.32·5-s − 3.41·7-s + 0.704·9-s − 3.42·11-s + 1.91·13-s + 2.54·15-s + 4.59·17-s − 3.32·19-s + 6.56·21-s + 2.08·23-s − 3.25·25-s + 4.41·27-s + 0.795·29-s − 9.99·31-s + 6.59·33-s + 4.50·35-s + 2.08·37-s − 3.68·39-s − 0.591·41-s + 8.49·43-s − 0.930·45-s + 1.41·47-s + 4.63·49-s − 8.84·51-s + 5.53·53-s + 4.52·55-s + ⋯ |
L(s) = 1 | − 1.11·3-s − 0.590·5-s − 1.28·7-s + 0.234·9-s − 1.03·11-s + 0.531·13-s + 0.656·15-s + 1.11·17-s − 0.763·19-s + 1.43·21-s + 0.435·23-s − 0.651·25-s + 0.850·27-s + 0.147·29-s − 1.79·31-s + 1.14·33-s + 0.761·35-s + 0.342·37-s − 0.590·39-s − 0.0923·41-s + 1.29·43-s − 0.138·45-s + 0.207·47-s + 0.661·49-s − 1.23·51-s + 0.759·53-s + 0.609·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 + 1.92T + 3T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 29 | \( 1 - 0.795T + 29T^{2} \) |
| 31 | \( 1 + 9.99T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 + 0.591T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 + 3.25T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 71 | \( 1 + 2.67T + 71T^{2} \) |
| 73 | \( 1 + 3.87T + 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 0.475T + 97T^{2} \) |
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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
−7.39476360043045825692146542300, −6.72718976351233802811727638771, −5.88674163180476088910460459697, −5.67408834626095272152671946295, −4.79848406259037911343289681732, −3.81727977059363652949591259882, −3.27836596677642460311321893325, −2.30271678800860927775672777319, −0.802780537458308852777430296644, 0,
0.802780537458308852777430296644, 2.30271678800860927775672777319, 3.27836596677642460311321893325, 3.81727977059363652949591259882, 4.79848406259037911343289681732, 5.67408834626095272152671946295, 5.88674163180476088910460459697, 6.72718976351233802811727638771, 7.39476360043045825692146542300