| L(s) = 1 | + 2·2-s − 0.313·3-s + 4·4-s − 0.626·6-s − 8.77·7-s + 8·8-s − 26.9·9-s − 55.0·11-s − 1.25·12-s + 15.2·13-s − 17.5·14-s + 16·16-s − 48.5·17-s − 53.8·18-s + 109.·19-s + 2.75·21-s − 110.·22-s − 113.·23-s − 2.50·24-s + 30.4·26-s + 16.8·27-s − 35.1·28-s + 169.·29-s + 246.·31-s + 32·32-s + 17.2·33-s − 97.0·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0603·3-s + 0.5·4-s − 0.0426·6-s − 0.473·7-s + 0.353·8-s − 0.996·9-s − 1.50·11-s − 0.0301·12-s + 0.324·13-s − 0.335·14-s + 0.250·16-s − 0.692·17-s − 0.704·18-s + 1.32·19-s + 0.0285·21-s − 1.06·22-s − 1.02·23-s − 0.0213·24-s + 0.229·26-s + 0.120·27-s − 0.236·28-s + 1.08·29-s + 1.42·31-s + 0.176·32-s + 0.0909·33-s − 0.489·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.393916667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393916667\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.313T + 27T^{2} \) |
| 7 | \( 1 + 8.77T + 343T^{2} \) |
| 11 | \( 1 + 55.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 246.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 367.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 318.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 9.34T + 1.48e5T^{2} \) |
| 59 | \( 1 + 167.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 425.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 241.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 380.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 622.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 7.92T + 4.93e5T^{2} \) |
| 83 | \( 1 + 484.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.59e3T + 9.12e5T^{2} \) |
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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
−9.445914547683587839212707965143, −8.252061082323013312073763046941, −7.78982022018323772317527050581, −6.59576358933502868865301642996, −5.88258727162519738624377997367, −5.17187368531606052200967971822, −4.19465714605442694507798410450, −2.95166130870093972454772646203, −2.50227590064681484137741610720, −0.68059475045234877564356964650,
0.68059475045234877564356964650, 2.50227590064681484137741610720, 2.95166130870093972454772646203, 4.19465714605442694507798410450, 5.17187368531606052200967971822, 5.88258727162519738624377997367, 6.59576358933502868865301642996, 7.78982022018323772317527050581, 8.252061082323013312073763046941, 9.445914547683587839212707965143
