L(s) = 1 | + 5-s + 2·7-s − 11-s + 13-s + 2·17-s + 8·23-s + 25-s + 8·29-s − 6·31-s + 2·35-s − 10·41-s + 8·43-s − 3·49-s + 4·53-s − 55-s − 8·59-s − 10·61-s + 65-s + 12·67-s − 8·71-s − 2·73-s − 2·77-s − 16·83-s + 2·85-s − 18·89-s + 2·91-s − 12·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.301·11-s + 0.277·13-s + 0.485·17-s + 1.66·23-s + 1/5·25-s + 1.48·29-s − 1.07·31-s + 0.338·35-s − 1.56·41-s + 1.21·43-s − 3/7·49-s + 0.549·53-s − 0.134·55-s − 1.04·59-s − 1.28·61-s + 0.124·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s − 0.227·77-s − 1.75·83-s + 0.216·85-s − 1.90·89-s + 0.209·91-s − 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.08106093276753, −13.53674155490135, −12.93107045379630, −12.61541228217781, −12.01342555830517, −11.49658390818425, −10.87462285557412, −10.72801784859850, −10.02908313465429, −9.577908198284105, −8.894476042791216, −8.600600392259570, −8.018903695762566, −7.461227305347879, −6.914335898161086, −6.481057558728591, −5.597810647341659, −5.438415371753781, −4.700701054844403, −4.348981352654101, −3.399688962900718, −2.955009070039592, −2.318670598683042, −1.422239588023986, −1.154918110577102, 0,
1.154918110577102, 1.422239588023986, 2.318670598683042, 2.955009070039592, 3.399688962900718, 4.348981352654101, 4.700701054844403, 5.438415371753781, 5.597810647341659, 6.481057558728591, 6.914335898161086, 7.461227305347879, 8.018903695762566, 8.600600392259570, 8.894476042791216, 9.577908198284105, 10.02908313465429, 10.72801784859850, 10.87462285557412, 11.49658390818425, 12.01342555830517, 12.61541228217781, 12.93107045379630, 13.53674155490135, 14.08106093276753