L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s − 3.62·13-s + 14-s + 16-s + 4.20·17-s − 8.15·19-s − 20-s + 22-s − 0.897·23-s + 25-s + 3.62·26-s − 28-s + 7.30·29-s − 3.42·31-s − 32-s − 4.20·34-s + 35-s − 1.10·37-s + 8.15·38-s + 40-s − 12.3·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 1.00·13-s + 0.267·14-s + 0.250·16-s + 1.01·17-s − 1.87·19-s − 0.223·20-s + 0.213·22-s − 0.187·23-s + 0.200·25-s + 0.711·26-s − 0.188·28-s + 1.35·29-s − 0.614·31-s − 0.176·32-s − 0.721·34-s + 0.169·35-s − 0.181·37-s + 1.32·38-s + 0.158·40-s − 1.92·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6522434699\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6522434699\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 + 8.15T + 19T^{2} \) |
| 23 | \( 1 + 0.897T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 1.15T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 + 8.41T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 0.205T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 2.20T + 83T^{2} \) |
| 89 | \( 1 - 7.45T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−8.037004480467048385745426043993, −7.37452533853338466293372674717, −6.67107685722981853069573175068, −6.07081673255192813437186709971, −5.10296802760846495978631745106, −4.38234936626767972937740745985, −3.41034116126752265772606894938, −2.65575464892903192954408223640, −1.76441068357714542664903479573, −0.44289500760624363717499421018,
0.44289500760624363717499421018, 1.76441068357714542664903479573, 2.65575464892903192954408223640, 3.41034116126752265772606894938, 4.38234936626767972937740745985, 5.10296802760846495978631745106, 6.07081673255192813437186709971, 6.67107685722981853069573175068, 7.37452533853338466293372674717, 8.037004480467048385745426043993