L(s) = 1 | − 0.0763·3-s + 1.17·5-s − 0.469·7-s − 2.99·9-s + 5.74·11-s − 1.85·13-s − 0.0895·15-s + 5.22·17-s − 2.12·19-s + 0.0358·21-s − 0.171·23-s − 3.62·25-s + 0.457·27-s − 6.19·29-s + 0.396·31-s − 0.438·33-s − 0.550·35-s − 8.17·37-s + 0.141·39-s − 12.4·41-s + 4.97·43-s − 3.51·45-s + 0.521·47-s − 6.77·49-s − 0.399·51-s + 8.76·53-s + 6.73·55-s + ⋯ |
L(s) = 1 | − 0.0440·3-s + 0.524·5-s − 0.177·7-s − 0.998·9-s + 1.73·11-s − 0.515·13-s − 0.0231·15-s + 1.26·17-s − 0.487·19-s + 0.00782·21-s − 0.0358·23-s − 0.724·25-s + 0.0880·27-s − 1.14·29-s + 0.0711·31-s − 0.0763·33-s − 0.0930·35-s − 1.34·37-s + 0.0227·39-s − 1.94·41-s + 0.759·43-s − 0.523·45-s + 0.0760·47-s − 0.968·49-s − 0.0559·51-s + 1.20·53-s + 0.908·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 0.0763T + 3T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 7 | \( 1 + 0.469T + 7T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 + 0.171T + 23T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 - 0.396T + 31T^{2} \) |
| 37 | \( 1 + 8.17T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 4.97T + 43T^{2} \) |
| 47 | \( 1 - 0.521T + 47T^{2} \) |
| 53 | \( 1 - 8.76T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 + 8.42T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 4.60T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 + 5.97T + 83T^{2} \) |
| 89 | \( 1 + 4.25T + 89T^{2} \) |
| 97 | \( 1 + 2.64T + 97T^{2} \) |
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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
−7.41316063409238546946471360434, −6.73091075530571326467004479418, −6.00661499864123596420927921796, −5.59476336404322237565233754429, −4.72931879358692407899781319625, −3.70915906663825062570534642815, −3.26142099972572349957497134530, −2.11173797827183793348824636030, −1.37955790277592905801675394156, 0,
1.37955790277592905801675394156, 2.11173797827183793348824636030, 3.26142099972572349957497134530, 3.70915906663825062570534642815, 4.72931879358692407899781319625, 5.59476336404322237565233754429, 6.00661499864123596420927921796, 6.73091075530571326467004479418, 7.41316063409238546946471360434