| L(s) = 1 | + 2-s + 4-s + 8-s + 4·11-s + 13-s + 16-s + 2·17-s + 4·22-s − 4·23-s + 26-s − 29-s − 8·31-s + 32-s + 2·34-s + 10·37-s + 2·41-s + 12·43-s + 4·44-s − 4·46-s − 7·49-s + 52-s − 10·53-s − 58-s − 12·59-s − 14·61-s − 8·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s + 0.277·13-s + 1/4·16-s + 0.485·17-s + 0.852·22-s − 0.834·23-s + 0.196·26-s − 0.185·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 1.64·37-s + 0.312·41-s + 1.82·43-s + 0.603·44-s − 0.589·46-s − 49-s + 0.138·52-s − 1.37·53-s − 0.131·58-s − 1.56·59-s − 1.79·61-s − 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.844898686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.844898686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.22481344839377, −12.66044038224465, −12.23027153147413, −12.09483725892298, −11.18795979296636, −10.99888052158013, −10.70058269876839, −9.615985741513720, −9.489533433466502, −9.166168646370623, −8.163601799530557, −7.938403697916910, −7.390253388981184, −6.761653933237938, −6.239039968788227, −5.936510992366921, −5.410967113618086, −4.617659373516706, −4.284954239032459, −3.680880283211714, −3.291711944676617, −2.561940254437370, −1.849644131647015, −1.364867254528960, −0.5647549886127564,
0.5647549886127564, 1.364867254528960, 1.849644131647015, 2.561940254437370, 3.291711944676617, 3.680880283211714, 4.284954239032459, 4.617659373516706, 5.410967113618086, 5.936510992366921, 6.239039968788227, 6.761653933237938, 7.390253388981184, 7.938403697916910, 8.163601799530557, 9.166168646370623, 9.489533433466502, 9.615985741513720, 10.70058269876839, 10.99888052158013, 11.18795979296636, 12.09483725892298, 12.23027153147413, 12.66044038224465, 13.22481344839377
