L(s) = 1 | − 3·9-s + 4·11-s − 2·13-s − 6·17-s + 8·19-s − 6·29-s − 8·31-s − 2·37-s − 2·41-s − 4·43-s − 8·47-s + 6·53-s − 6·61-s − 4·67-s − 8·71-s + 10·73-s + 16·79-s + 9·81-s − 8·83-s + 6·89-s − 6·97-s − 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 9-s + 1.20·11-s − 0.554·13-s − 1.45·17-s + 1.83·19-s − 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 0.824·53-s − 0.768·61-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 1.80·79-s + 81-s − 0.878·83-s + 0.635·89-s − 0.609·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.097799050\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.097799050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.90676502767873, −13.62968640047139, −13.16063179391824, −12.37230686495849, −11.97369596072002, −11.52412358979771, −11.11765392853476, −10.71561518859567, −9.733876620159844, −9.513392350663991, −8.955174425121569, −8.634908193746423, −7.831897991060344, −7.352443485967472, −6.804731723485505, −6.340603132579929, −5.608361669922616, −5.221713284375482, −4.621510223571975, −3.780149620091144, −3.446211831243960, −2.719549687489290, −1.973700893256161, −1.393419472809358, −0.3363903945294763,
0.3363903945294763, 1.393419472809358, 1.973700893256161, 2.719549687489290, 3.446211831243960, 3.780149620091144, 4.621510223571975, 5.221713284375482, 5.608361669922616, 6.340603132579929, 6.804731723485505, 7.352443485967472, 7.831897991060344, 8.634908193746423, 8.955174425121569, 9.513392350663991, 9.733876620159844, 10.71561518859567, 11.11765392853476, 11.52412358979771, 11.97369596072002, 12.37230686495849, 13.16063179391824, 13.62968640047139, 13.90676502767873