| L(s) = 1 | − 2·4-s − 2·5-s + 2·7-s + 11-s − 5·13-s + 4·16-s + 3·17-s + 4·20-s − 6·23-s − 25-s − 4·28-s − 10·29-s − 10·31-s − 4·35-s − 10·41-s − 6·43-s − 2·44-s + 2·47-s − 3·49-s + 10·52-s + 5·53-s − 2·55-s − 5·59-s − 8·64-s + 10·65-s − 10·67-s − 6·68-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s + 0.755·7-s + 0.301·11-s − 1.38·13-s + 16-s + 0.727·17-s + 0.894·20-s − 1.25·23-s − 1/5·25-s − 0.755·28-s − 1.85·29-s − 1.79·31-s − 0.676·35-s − 1.56·41-s − 0.914·43-s − 0.301·44-s + 0.291·47-s − 3/7·49-s + 1.38·52-s + 0.686·53-s − 0.269·55-s − 0.650·59-s − 64-s + 1.24·65-s − 1.22·67-s − 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 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
−15.08108849017373, −14.82383785803327, −14.64464358051832, −13.85968252368726, −13.47294539169470, −12.70693533434723, −12.28327321649341, −11.80313763839510, −11.42221824635763, −10.62736015545344, −10.07545143108216, −9.543324890059704, −9.068231957261125, −8.390754563255794, −7.799743791808884, −7.589081260679571, −6.978157985614531, −5.911589796961541, −5.356908605672917, −4.941611894286347, −4.185796882856437, −3.781575349351974, −3.203753752336370, −2.023071304169786, −1.470913199991305, 0, 0,
1.470913199991305, 2.023071304169786, 3.203753752336370, 3.781575349351974, 4.185796882856437, 4.941611894286347, 5.356908605672917, 5.911589796961541, 6.978157985614531, 7.589081260679571, 7.799743791808884, 8.390754563255794, 9.068231957261125, 9.543324890059704, 10.07545143108216, 10.62736015545344, 11.42221824635763, 11.80313763839510, 12.28327321649341, 12.70693533434723, 13.47294539169470, 13.85968252368726, 14.64464358051832, 14.82383785803327, 15.08108849017373
