| L(s) = 1 | + 3-s − 3·7-s + 9-s − 5·11-s + 13-s − 5·17-s − 4·19-s − 3·21-s − 2·23-s + 27-s − 9·29-s − 3·31-s − 5·33-s − 10·37-s + 39-s − 12·41-s − 2·43-s − 9·47-s + 2·49-s − 5·51-s + 9·53-s − 4·57-s − 3·59-s − 7·61-s − 3·63-s − 9·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s − 1.21·17-s − 0.917·19-s − 0.654·21-s − 0.417·23-s + 0.192·27-s − 1.67·29-s − 0.538·31-s − 0.870·33-s − 1.64·37-s + 0.160·39-s − 1.87·41-s − 0.304·43-s − 1.31·47-s + 2/7·49-s − 0.700·51-s + 1.23·53-s − 0.529·57-s − 0.390·59-s − 0.896·61-s − 0.377·63-s − 1.09·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.55401444528221, −15.14303745885243, −14.77505075034140, −13.70184004009179, −13.53607620140488, −13.00390442010677, −12.74024431217843, −12.00491930885621, −11.27730756367304, −10.61185218847557, −10.32207164709397, −9.719625436730620, −9.073371468431459, −8.586807806397368, −8.144820431384140, −7.288766358460208, −6.965159047998572, −6.247419426491829, −5.661924102076261, −4.941725643337696, −4.286891903059386, −3.377406907995885, −3.207095957677966, −2.127653848570824, −1.844308605739677, 0, 0,
1.844308605739677, 2.127653848570824, 3.207095957677966, 3.377406907995885, 4.286891903059386, 4.941725643337696, 5.661924102076261, 6.247419426491829, 6.965159047998572, 7.288766358460208, 8.144820431384140, 8.586807806397368, 9.073371468431459, 9.719625436730620, 10.32207164709397, 10.61185218847557, 11.27730756367304, 12.00491930885621, 12.74024431217843, 13.00390442010677, 13.53607620140488, 13.70184004009179, 14.77505075034140, 15.14303745885243, 15.55401444528221
