L(s) = 1 | + 2-s + 4-s + 8-s + 11-s + 2.82·13-s + 16-s + 2.82·17-s − 5.65·19-s + 22-s − 8·23-s − 5·25-s + 2.82·26-s − 2·29-s − 8.48·31-s + 32-s + 2.82·34-s + 2·37-s − 5.65·38-s + 2.82·41-s − 4·43-s + 44-s − 8·46-s + 2.82·47-s − 5·50-s + 2.82·52-s − 14·53-s − 2·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s + 0.301·11-s + 0.784·13-s + 0.250·16-s + 0.685·17-s − 1.29·19-s + 0.213·22-s − 1.66·23-s − 25-s + 0.554·26-s − 0.371·29-s − 1.52·31-s + 0.176·32-s + 0.485·34-s + 0.328·37-s − 0.917·38-s + 0.441·41-s − 0.609·43-s + 0.150·44-s − 1.17·46-s + 0.412·47-s − 0.707·50-s + 0.392·52-s − 1.92·53-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.26291126010817866608730322856, −6.45379480453288463630673707805, −5.91945577831179393514408992648, −5.44412349960301715849966381951, −4.31742103928807866958260303629, −3.94894170634214286509505704383, −3.23199485372711231632169723570, −2.12973997053598942616145080949, −1.53611668743012339964030891379, 0,
1.53611668743012339964030891379, 2.12973997053598942616145080949, 3.23199485372711231632169723570, 3.94894170634214286509505704383, 4.31742103928807866958260303629, 5.44412349960301715849966381951, 5.91945577831179393514408992648, 6.45379480453288463630673707805, 7.26291126010817866608730322856