| L(s) = 1 | − 3.56·2-s + 4.72·4-s + 20.6·5-s − 5.24·7-s + 11.6·8-s − 73.6·10-s + 5.21·11-s − 14.8·13-s + 18.7·14-s − 79.4·16-s − 17·17-s + 26.2·19-s + 97.5·20-s − 18.5·22-s + 165.·23-s + 300.·25-s + 52.8·26-s − 24.7·28-s − 42.3·29-s + 263.·31-s + 190.·32-s + 60.6·34-s − 108.·35-s − 322.·37-s − 93.6·38-s + 240.·40-s + 321.·41-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.591·4-s + 1.84·5-s − 0.283·7-s + 0.515·8-s − 2.32·10-s + 0.142·11-s − 0.315·13-s + 0.357·14-s − 1.24·16-s − 0.242·17-s + 0.317·19-s + 1.09·20-s − 0.180·22-s + 1.49·23-s + 2.40·25-s + 0.398·26-s − 0.167·28-s − 0.271·29-s + 1.52·31-s + 1.05·32-s + 0.305·34-s − 0.522·35-s − 1.43·37-s − 0.399·38-s + 0.951·40-s + 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.156750210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156750210\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 3.56T + 8T^{2} \) |
| 5 | \( 1 - 20.6T + 125T^{2} \) |
| 7 | \( 1 + 5.24T + 343T^{2} \) |
| 11 | \( 1 - 5.21T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.8T + 2.19e3T^{2} \) |
| 19 | \( 1 - 26.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 42.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 322.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 385.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 192.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 241.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 933.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 869.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 102.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 298.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 666.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.35e3T + 9.12e5T^{2} \) |
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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
−12.61825142944827577183203785288, −10.97289370472511688948647731685, −10.17828974970501294919493201837, −9.391305296833257148221719960173, −8.821769211335353225351863345315, −7.29381656739139693234283352364, −6.24554186892528550864988959631, −4.94720697761208684389158774293, −2.52304543565308784976037146098, −1.14911976625303827165170692907,
1.14911976625303827165170692907, 2.52304543565308784976037146098, 4.94720697761208684389158774293, 6.24554186892528550864988959631, 7.29381656739139693234283352364, 8.821769211335353225351863345315, 9.391305296833257148221719960173, 10.17828974970501294919493201837, 10.97289370472511688948647731685, 12.61825142944827577183203785288
