L(s) = 1 | − 2-s + 4-s − 8-s + 2·11-s + 4·13-s + 16-s + 2·17-s + 8·19-s − 2·22-s + 8·23-s − 5·25-s − 4·26-s + 6·29-s + 8·31-s − 32-s − 2·34-s − 6·37-s − 8·38-s − 10·41-s − 12·43-s + 2·44-s − 8·46-s − 7·49-s + 5·50-s + 4·52-s − 6·53-s − 6·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.603·11-s + 1.10·13-s + 1/4·16-s + 0.485·17-s + 1.83·19-s − 0.426·22-s + 1.66·23-s − 25-s − 0.784·26-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.986·37-s − 1.29·38-s − 1.56·41-s − 1.82·43-s + 0.301·44-s − 1.17·46-s − 49-s + 0.707·50-s + 0.554·52-s − 0.824·53-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.700013389\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.700013389\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.415144397830555473692055094464, −7.932385790203622441424541840968, −6.86855674915499663081512023480, −6.56503396722632913734144265292, −5.50611330338385692726488796754, −4.82509318971821511998202881621, −3.48355474067943631761989991423, −3.10547844973673522184709026899, −1.61300127724440495001158810244, −0.917290078705867187056662172773,
0.917290078705867187056662172773, 1.61300127724440495001158810244, 3.10547844973673522184709026899, 3.48355474067943631761989991423, 4.82509318971821511998202881621, 5.50611330338385692726488796754, 6.56503396722632913734144265292, 6.86855674915499663081512023480, 7.932385790203622441424541840968, 8.415144397830555473692055094464