| L(s) = 1 | − 1.33·2-s − 6.23·4-s − 10.6·7-s + 18.9·8-s + 11.2·11-s + 2.74·13-s + 14.2·14-s + 24.6·16-s − 29.5·17-s − 31.1·19-s − 14.9·22-s + 116.·23-s − 3.64·26-s + 66.6·28-s + 108.·29-s + 70.7·31-s − 184.·32-s + 39.3·34-s + 282.·37-s + 41.3·38-s − 425.·41-s + 312.·43-s − 70.1·44-s − 155.·46-s − 193.·47-s − 228.·49-s − 17.0·52-s + ⋯ |
| L(s) = 1 | − 0.470·2-s − 0.778·4-s − 0.577·7-s + 0.836·8-s + 0.308·11-s + 0.0584·13-s + 0.271·14-s + 0.385·16-s − 0.421·17-s − 0.375·19-s − 0.145·22-s + 1.06·23-s − 0.0274·26-s + 0.449·28-s + 0.694·29-s + 0.410·31-s − 1.01·32-s + 0.198·34-s + 1.25·37-s + 0.176·38-s − 1.62·41-s + 1.10·43-s − 0.240·44-s − 0.498·46-s − 0.600·47-s − 0.666·49-s − 0.0455·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.33T + 8T^{2} \) |
| 7 | \( 1 + 10.6T + 343T^{2} \) |
| 11 | \( 1 - 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.74T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 70.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 282.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 425.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 312.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 193.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 103.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 494.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 424.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 586.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 302.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 525.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 25.6T + 9.12e5T^{2} \) |
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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
−9.575439202245713103097864129546, −8.888056111064116583324809754175, −8.143660034351238330894314015371, −7.07881779161021608294264061851, −6.17555228664415708446372483124, −4.96043870629066421312157800751, −4.11178819592755729691750946418, −2.90948434843223767084979142461, −1.26709928904406110050944629086, 0,
1.26709928904406110050944629086, 2.90948434843223767084979142461, 4.11178819592755729691750946418, 4.96043870629066421312157800751, 6.17555228664415708446372483124, 7.07881779161021608294264061851, 8.143660034351238330894314015371, 8.888056111064116583324809754175, 9.575439202245713103097864129546
