L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s + 2·11-s + 5·13-s − 14-s − 16-s + 4·17-s − 2·22-s + 4·23-s − 5·25-s − 5·26-s − 28-s + 8·29-s − 3·31-s − 5·32-s − 4·34-s + 3·37-s + 12·41-s − 43-s − 2·44-s − 4·46-s + 6·47-s − 6·49-s + 5·50-s − 5·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s + 0.603·11-s + 1.38·13-s − 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.426·22-s + 0.834·23-s − 25-s − 0.980·26-s − 0.188·28-s + 1.48·29-s − 0.538·31-s − 0.883·32-s − 0.685·34-s + 0.493·37-s + 1.87·41-s − 0.152·43-s − 0.301·44-s − 0.589·46-s + 0.875·47-s − 6/7·49-s + 0.707·50-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.390320045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.390320045\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.768911518346064668164493132159, −7.950432921708495475286177817984, −7.51985913573493673264860332316, −6.37928566570466283360048966408, −5.72108152384040525575330835952, −4.69567777034077115354325976863, −4.02601075420258188158144308698, −3.10731246257957653119123158491, −1.59046227323328773833712195788, −0.888187312996876518569196306142,
0.888187312996876518569196306142, 1.59046227323328773833712195788, 3.10731246257957653119123158491, 4.02601075420258188158144308698, 4.69567777034077115354325976863, 5.72108152384040525575330835952, 6.37928566570466283360048966408, 7.51985913573493673264860332316, 7.950432921708495475286177817984, 8.768911518346064668164493132159