L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 4·14-s + 16-s − 4·17-s + 8·19-s + 20-s + 23-s + 25-s − 4·28-s + 2·29-s + 4·31-s − 32-s + 4·34-s − 4·35-s − 2·37-s − 8·38-s − 40-s − 8·41-s − 4·43-s − 46-s − 8·47-s + 9·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 1.06·14-s + 1/4·16-s − 0.970·17-s + 1.83·19-s + 0.223·20-s + 0.208·23-s + 1/5·25-s − 0.755·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.676·35-s − 0.328·37-s − 1.29·38-s − 0.158·40-s − 1.24·41-s − 0.609·43-s − 0.147·46-s − 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.933171196373164436028073628791, −8.047403151846088508381664903301, −7.02078577529983508395697282954, −6.58094930913824246215386749564, −5.78281516922020646818108599784, −4.77008429136827408025826825802, −3.35966149519111135869782665267, −2.82354330440148796947542327122, −1.45402989302954842587531760308, 0,
1.45402989302954842587531760308, 2.82354330440148796947542327122, 3.35966149519111135869782665267, 4.77008429136827408025826825802, 5.78281516922020646818108599784, 6.58094930913824246215386749564, 7.02078577529983508395697282954, 8.047403151846088508381664903301, 8.933171196373164436028073628791