L(s) = 1 | + 7-s − 6·13-s + 4·17-s + 19-s + 2·23-s − 5·25-s + 6·29-s − 3·31-s + 7·37-s + 2·41-s − 4·43-s + 6·47-s − 6·49-s + 2·53-s − 10·59-s + 61-s − 3·67-s + 4·71-s + 3·73-s − 5·79-s + 14·83-s − 16·89-s − 6·91-s + 13·97-s + 18·101-s + 15·103-s − 10·107-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 1.66·13-s + 0.970·17-s + 0.229·19-s + 0.417·23-s − 25-s + 1.11·29-s − 0.538·31-s + 1.15·37-s + 0.312·41-s − 0.609·43-s + 0.875·47-s − 6/7·49-s + 0.274·53-s − 1.30·59-s + 0.128·61-s − 0.366·67-s + 0.474·71-s + 0.351·73-s − 0.562·79-s + 1.53·83-s − 1.69·89-s − 0.628·91-s + 1.31·97-s + 1.79·101-s + 1.47·103-s − 0.966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.866250715\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.866250715\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.65651185205430341336090215657, −7.30349526414123229608598331782, −6.37543367109650847006329019924, −5.63803704368390433919464015794, −4.93901642234231043139140289386, −4.41416207821389809574569870271, −3.38638097901984956033745578771, −2.64261050151664183829261569136, −1.79828342447873620842600937396, −0.66132008878690134841942812743,
0.66132008878690134841942812743, 1.79828342447873620842600937396, 2.64261050151664183829261569136, 3.38638097901984956033745578771, 4.41416207821389809574569870271, 4.93901642234231043139140289386, 5.63803704368390433919464015794, 6.37543367109650847006329019924, 7.30349526414123229608598331782, 7.65651185205430341336090215657