L(s) = 1 | + 4·5-s − 3·9-s − 4·13-s − 8·17-s + 11·25-s − 10·29-s − 2·37-s − 8·41-s − 12·45-s − 14·53-s − 12·61-s − 16·65-s + 16·73-s + 9·81-s − 32·85-s + 16·89-s + 8·97-s + 20·101-s + 6·109-s − 14·113-s + 12·117-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 9-s − 1.10·13-s − 1.94·17-s + 11/5·25-s − 1.85·29-s − 0.328·37-s − 1.24·41-s − 1.78·45-s − 1.92·53-s − 1.53·61-s − 1.98·65-s + 1.87·73-s + 81-s − 3.47·85-s + 1.69·89-s + 0.812·97-s + 1.99·101-s + 0.574·109-s − 1.31·113-s + 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.537568760765362109873978216969, −7.49044587751322728460045665733, −6.56423418036942605468335994950, −6.11930935179889663595804173654, −5.23150515102433129890608359150, −4.75116456994544894201449242731, −3.30936901781950835016652147992, −2.30242226022330946580243876877, −1.88143167409037862528166102555, 0,
1.88143167409037862528166102555, 2.30242226022330946580243876877, 3.30936901781950835016652147992, 4.75116456994544894201449242731, 5.23150515102433129890608359150, 6.11930935179889663595804173654, 6.56423418036942605468335994950, 7.49044587751322728460045665733, 8.537568760765362109873978216969