| L(s) = 1 | − 3-s + 7-s − 2·9-s − 6·11-s + 5·13-s + 3·17-s − 21-s − 3·23-s − 5·25-s + 5·27-s + 9·29-s − 4·31-s + 6·33-s + 2·37-s − 5·39-s + 8·43-s − 6·49-s − 3·51-s − 3·53-s − 9·59-s + 10·61-s − 2·63-s − 5·67-s + 3·69-s − 6·71-s − 7·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.80·11-s + 1.38·13-s + 0.727·17-s − 0.218·21-s − 0.625·23-s − 25-s + 0.962·27-s + 1.67·29-s − 0.718·31-s + 1.04·33-s + 0.328·37-s − 0.800·39-s + 1.21·43-s − 6/7·49-s − 0.420·51-s − 0.412·53-s − 1.17·59-s + 1.28·61-s − 0.251·63-s − 0.610·67-s + 0.361·69-s − 0.712·71-s − 0.819·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.71056411360796, −15.54428165346785, −14.41245770721739, −14.25937506093475, −13.52723115327774, −13.06654074555751, −12.48144148764082, −11.84215311301444, −11.36099953147452, −10.84426646877209, −10.35929918921294, −9.935983409060091, −8.958638266035974, −8.444417260206257, −7.895349544747219, −7.553875984406283, −6.516781767740537, −5.841205525554877, −5.683093376605319, −4.898122828592201, −4.280449725338770, −3.316688471103953, −2.799063666364987, −1.911534164762365, −0.9323031052651675, 0,
0.9323031052651675, 1.911534164762365, 2.799063666364987, 3.316688471103953, 4.280449725338770, 4.898122828592201, 5.683093376605319, 5.841205525554877, 6.516781767740537, 7.553875984406283, 7.895349544747219, 8.444417260206257, 8.958638266035974, 9.935983409060091, 10.35929918921294, 10.84426646877209, 11.36099953147452, 11.84215311301444, 12.48144148764082, 13.06654074555751, 13.52723115327774, 14.25937506093475, 14.41245770721739, 15.54428165346785, 15.71056411360796
