L(s) = 1 | − 2·3-s + 9-s + 2·13-s − 6·17-s + 4·19-s − 6·23-s + 4·27-s + 6·29-s + 4·31-s − 2·37-s − 4·39-s − 6·41-s + 10·43-s − 6·47-s + 12·51-s + 6·53-s − 8·57-s − 12·59-s − 2·61-s − 2·67-s + 12·69-s − 12·71-s + 2·73-s + 8·79-s − 11·81-s + 6·83-s − 12·87-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s − 0.875·47-s + 1.68·51-s + 0.824·53-s − 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.244·67-s + 1.44·69-s − 1.42·71-s + 0.234·73-s + 0.900·79-s − 1.22·81-s + 0.658·83-s − 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.88419219340670143835762506406, −6.97551092000674168681064640184, −6.27820172149121365823505974667, −5.90704167986561784739502061162, −4.92404472756688457136741701293, −4.42854867231847186766757062918, −3.37783318018363163987837300608, −2.32979843560722233251320566800, −1.13781969259127232409360054598, 0,
1.13781969259127232409360054598, 2.32979843560722233251320566800, 3.37783318018363163987837300608, 4.42854867231847186766757062918, 4.92404472756688457136741701293, 5.90704167986561784739502061162, 6.27820172149121365823505974667, 6.97551092000674168681064640184, 7.88419219340670143835762506406