| L(s) = 1 | + 2·2-s + 8.20·3-s + 4·4-s + 16.4·6-s + 8·8-s + 40.3·9-s + 7.21·11-s + 32.8·12-s − 15.3·13-s + 16·16-s + 24.0·17-s + 80.6·18-s − 8.88·19-s + 14.4·22-s + 195.·23-s + 65.6·24-s − 30.6·26-s + 109.·27-s + 202.·29-s + 306.·31-s + 32·32-s + 59.2·33-s + 48.0·34-s + 161.·36-s − 147.·37-s − 17.7·38-s − 125.·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s + 1.11·6-s + 0.353·8-s + 1.49·9-s + 0.197·11-s + 0.789·12-s − 0.326·13-s + 0.250·16-s + 0.342·17-s + 1.05·18-s − 0.107·19-s + 0.139·22-s + 1.77·23-s + 0.558·24-s − 0.230·26-s + 0.780·27-s + 1.29·29-s + 1.77·31-s + 0.176·32-s + 0.312·33-s + 0.242·34-s + 0.746·36-s − 0.655·37-s − 0.0758·38-s − 0.515·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.602437546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.602437546\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 8.20T + 27T^{2} \) |
| 11 | \( 1 - 7.21T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 8.88T + 6.85e3T^{2} \) |
| 23 | \( 1 - 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 306.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 461.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 70.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 207.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 467.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 162.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 496.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 595.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 684.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 32.2T + 9.12e5T^{2} \) |
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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
−8.470877290648377994454239600038, −7.972090369125295850610237923507, −6.98735700615810312039075605326, −6.52461243357183932793997016273, −5.14436283811425074316169256321, −4.56195529892025514849667371818, −3.45275448510175860899660476684, −3.01353482426404618162934642700, −2.14024722493498776685806398849, −1.06925736565765436069987403862,
1.06925736565765436069987403862, 2.14024722493498776685806398849, 3.01353482426404618162934642700, 3.45275448510175860899660476684, 4.56195529892025514849667371818, 5.14436283811425074316169256321, 6.52461243357183932793997016273, 6.98735700615810312039075605326, 7.972090369125295850610237923507, 8.470877290648377994454239600038
