L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 2·7-s + 9-s + 5·11-s − 2·12-s − 4·14-s − 4·16-s + 4·17-s + 2·18-s + 19-s + 2·21-s + 10·22-s − 27-s − 4·28-s + 2·29-s + 7·31-s − 8·32-s − 5·33-s + 8·34-s + 2·36-s − 10·37-s + 2·38-s − 41-s + 4·42-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 0.755·7-s + 1/3·9-s + 1.50·11-s − 0.577·12-s − 1.06·14-s − 16-s + 0.970·17-s + 0.471·18-s + 0.229·19-s + 0.436·21-s + 2.13·22-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + 1.25·31-s − 1.41·32-s − 0.870·33-s + 1.37·34-s + 1/3·36-s − 1.64·37-s + 0.324·38-s − 0.156·41-s + 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.137927502\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.137927502\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.53460924635303, −14.23192625439679, −13.81891208376330, −13.27868552011139, −12.55003754858688, −12.27971042368796, −11.91748998090086, −11.45545306199767, −10.70403682998307, −10.21097701036909, −9.378334811432948, −9.241701361560656, −8.370840294919443, −7.541931089477811, −6.902016006241705, −6.338413199174863, −6.173494282081050, −5.396561871156454, −4.889016304011531, −4.222620402538773, −3.648122790269284, −3.253645379406524, −2.441184503063257, −1.453709237636859, −0.6339460008454390,
0.6339460008454390, 1.453709237636859, 2.441184503063257, 3.253645379406524, 3.648122790269284, 4.222620402538773, 4.889016304011531, 5.396561871156454, 6.173494282081050, 6.338413199174863, 6.902016006241705, 7.541931089477811, 8.370840294919443, 9.241701361560656, 9.378334811432948, 10.21097701036909, 10.70403682998307, 11.45545306199767, 11.91748998090086, 12.27971042368796, 12.55003754858688, 13.27868552011139, 13.81891208376330, 14.23192625439679, 14.53460924635303