| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s + 7·13-s + 16-s + 4·17-s + 19-s − 20-s − 22-s − 23-s + 25-s − 7·26-s + 8·29-s + 6·31-s − 32-s − 4·34-s − 3·37-s − 38-s + 40-s − 9·41-s − 4·43-s + 44-s + 46-s + 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.94·13-s + 1/4·16-s + 0.970·17-s + 0.229·19-s − 0.223·20-s − 0.213·22-s − 0.208·23-s + 1/5·25-s − 1.37·26-s + 1.48·29-s + 1.07·31-s − 0.176·32-s − 0.685·34-s − 0.493·37-s − 0.162·38-s + 0.158·40-s − 1.40·41-s − 0.609·43-s + 0.150·44-s + 0.147·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516429299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516429299\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.366843341319179148775855671040, −7.907236327101361732706910422643, −6.88130541488254988978003636063, −6.36345589351212518270104025289, −5.59402547822408073304172506998, −4.55253543436526259589926890506, −3.58489035347965120593714647534, −3.02590450359383283650633276131, −1.60232588667144123438392019436, −0.836124942228002545895628930893,
0.836124942228002545895628930893, 1.60232588667144123438392019436, 3.02590450359383283650633276131, 3.58489035347965120593714647534, 4.55253543436526259589926890506, 5.59402547822408073304172506998, 6.36345589351212518270104025289, 6.88130541488254988978003636063, 7.907236327101361732706910422643, 8.366843341319179148775855671040
