| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 8-s + 6·9-s − 5·11-s − 3·12-s − 6·13-s + 16-s − 17-s − 6·18-s + 3·19-s + 5·22-s + 3·24-s + 6·26-s − 9·27-s − 6·29-s + 4·31-s − 32-s + 15·33-s + 34-s + 6·36-s − 8·37-s − 3·38-s + 18·39-s − 11·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.353·8-s + 2·9-s − 1.50·11-s − 0.866·12-s − 1.66·13-s + 1/4·16-s − 0.242·17-s − 1.41·18-s + 0.688·19-s + 1.06·22-s + 0.612·24-s + 1.17·26-s − 1.73·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 2.61·33-s + 0.171·34-s + 36-s − 1.31·37-s − 0.486·38-s + 2.88·39-s − 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2191963188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2191963188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.142352068910926817460090443423, −7.909616856504655698109693901245, −7.35159550652469145874551234062, −6.75922904616944609353713204114, −5.69551226567491875356313493124, −5.23317326717424642267182154953, −4.53392433226581882017134096766, −2.96910811860602576493917153273, −1.82578348423128801401226534071, −0.34762960201747218551587326811,
0.34762960201747218551587326811, 1.82578348423128801401226534071, 2.96910811860602576493917153273, 4.53392433226581882017134096766, 5.23317326717424642267182154953, 5.69551226567491875356313493124, 6.75922904616944609353713204114, 7.35159550652469145874551234062, 7.909616856504655698109693901245, 9.142352068910926817460090443423
