L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 3·11-s + 5·13-s − 14-s + 16-s + 4·17-s + 19-s + 3·22-s + 8·23-s + 5·26-s − 28-s + 10·29-s − 8·31-s + 32-s + 4·34-s − 4·37-s + 38-s + 9·41-s + 43-s + 3·44-s + 8·46-s − 13·47-s + 49-s + 5·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.904·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s + 0.639·22-s + 1.66·23-s + 0.980·26-s − 0.188·28-s + 1.85·29-s − 1.43·31-s + 0.176·32-s + 0.685·34-s − 0.657·37-s + 0.162·38-s + 1.40·41-s + 0.152·43-s + 0.452·44-s + 1.17·46-s − 1.89·47-s + 1/7·49-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.249078387\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.249078387\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.49019757965160692996851661275, −6.88677041850148224886646883720, −6.23924120618635645704166878786, −5.74282984034056484224628395951, −4.88809373850617859834368852290, −4.19678706039949504151862940841, −3.25802207275967235072877402307, −3.10673200403949152712395275166, −1.63512553018908657154421161148, −0.978032641728812393781087060780,
0.978032641728812393781087060780, 1.63512553018908657154421161148, 3.10673200403949152712395275166, 3.25802207275967235072877402307, 4.19678706039949504151862940841, 4.88809373850617859834368852290, 5.74282984034056484224628395951, 6.23924120618635645704166878786, 6.88677041850148224886646883720, 7.49019757965160692996851661275