| L(s) = 1 | + 26·7-s + 59·11-s + 28·13-s − 5·17-s + 109·19-s + 194·23-s + 32·29-s + 10·31-s − 198·37-s − 117·41-s + 388·43-s + 68·47-s + 333·49-s + 18·53-s − 392·59-s − 710·61-s − 253·67-s + 612·71-s − 549·73-s + 1.53e3·77-s + 414·79-s + 121·83-s + 81·89-s + 728·91-s − 1.50e3·97-s + 234·101-s − 1.17e3·103-s + ⋯ |
| L(s) = 1 | + 1.40·7-s + 1.61·11-s + 0.597·13-s − 0.0713·17-s + 1.31·19-s + 1.75·23-s + 0.204·29-s + 0.0579·31-s − 0.879·37-s − 0.445·41-s + 1.37·43-s + 0.211·47-s + 0.970·49-s + 0.0466·53-s − 0.864·59-s − 1.49·61-s − 0.461·67-s + 1.02·71-s − 0.880·73-s + 2.27·77-s + 0.589·79-s + 0.160·83-s + 0.0964·89-s + 0.838·91-s − 1.57·97-s + 0.230·101-s − 1.12·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.629545547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.629545547\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 59 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 5 T + p^{3} T^{2} \) |
| 19 | \( 1 - 109 T + p^{3} T^{2} \) |
| 23 | \( 1 - 194 T + p^{3} T^{2} \) |
| 29 | \( 1 - 32 T + p^{3} T^{2} \) |
| 31 | \( 1 - 10 T + p^{3} T^{2} \) |
| 37 | \( 1 + 198 T + p^{3} T^{2} \) |
| 41 | \( 1 + 117 T + p^{3} T^{2} \) |
| 43 | \( 1 - 388 T + p^{3} T^{2} \) |
| 47 | \( 1 - 68 T + p^{3} T^{2} \) |
| 53 | \( 1 - 18 T + p^{3} T^{2} \) |
| 59 | \( 1 + 392 T + p^{3} T^{2} \) |
| 61 | \( 1 + 710 T + p^{3} T^{2} \) |
| 67 | \( 1 + 253 T + p^{3} T^{2} \) |
| 71 | \( 1 - 612 T + p^{3} T^{2} \) |
| 73 | \( 1 + 549 T + p^{3} T^{2} \) |
| 79 | \( 1 - 414 T + p^{3} T^{2} \) |
| 83 | \( 1 - 121 T + p^{3} T^{2} \) |
| 89 | \( 1 - 81 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1502 T + p^{3} T^{2} \) |
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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
−9.016402008685341299567428707416, −8.165595893192104506133524963190, −7.33281950977616883589934137057, −6.62800091795564502724397936876, −5.58949410716182188273299028400, −4.81980952953748768002918376368, −3.99066961525529980363948395970, −3.00248605232219332947351637516, −1.54240166267660364600056979332, −1.06057056802044278057984653921,
1.06057056802044278057984653921, 1.54240166267660364600056979332, 3.00248605232219332947351637516, 3.99066961525529980363948395970, 4.81980952953748768002918376368, 5.58949410716182188273299028400, 6.62800091795564502724397936876, 7.33281950977616883589934137057, 8.165595893192104506133524963190, 9.016402008685341299567428707416
