| L(s) = 1 | + 7·3-s − 5·5-s − 7·7-s + 22·9-s + 9·11-s − 23·13-s − 35·15-s + 41·17-s + 34·19-s − 49·21-s + 6·23-s + 25·25-s − 35·27-s − 131·29-s − 4·31-s + 63·33-s + 35·35-s − 26·37-s − 161·39-s − 260·41-s − 190·43-s − 110·45-s − 167·47-s + 49·49-s + 287·51-s + 368·53-s − 45·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 0.447·5-s − 0.377·7-s + 0.814·9-s + 0.246·11-s − 0.490·13-s − 0.602·15-s + 0.584·17-s + 0.410·19-s − 0.509·21-s + 0.0543·23-s + 1/5·25-s − 0.249·27-s − 0.838·29-s − 0.0231·31-s + 0.332·33-s + 0.169·35-s − 0.115·37-s − 0.661·39-s − 0.990·41-s − 0.673·43-s − 0.364·45-s − 0.518·47-s + 1/7·49-s + 0.788·51-s + 0.953·53-s − 0.110·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 23 T + p^{3} T^{2} \) |
| 17 | \( 1 - 41 T + p^{3} T^{2} \) |
| 19 | \( 1 - 34 T + p^{3} T^{2} \) |
| 23 | \( 1 - 6 T + p^{3} T^{2} \) |
| 29 | \( 1 + 131 T + p^{3} T^{2} \) |
| 31 | \( 1 + 4 T + p^{3} T^{2} \) |
| 37 | \( 1 + 26 T + p^{3} T^{2} \) |
| 41 | \( 1 + 260 T + p^{3} T^{2} \) |
| 43 | \( 1 + 190 T + p^{3} T^{2} \) |
| 47 | \( 1 + 167 T + p^{3} T^{2} \) |
| 53 | \( 1 - 368 T + p^{3} T^{2} \) |
| 59 | \( 1 - 324 T + p^{3} T^{2} \) |
| 61 | \( 1 - 164 T + p^{3} T^{2} \) |
| 67 | \( 1 - 200 T + p^{3} T^{2} \) |
| 71 | \( 1 + 784 T + p^{3} T^{2} \) |
| 73 | \( 1 + 410 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1211 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1132 T + p^{3} T^{2} \) |
| 89 | \( 1 + 72 T + p^{3} T^{2} \) |
| 97 | \( 1 + 707 T + p^{3} T^{2} \) |
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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
−8.475977037886668322684086559009, −7.48481692501882319981493975593, −7.15526235699589602664725007762, −5.96786533276493563587428428485, −4.98537747832813261903967458496, −3.90064479321008421013119269228, −3.32436871777128967980853293130, −2.52291330431898173709159151868, −1.44564222440192788040498055648, 0,
1.44564222440192788040498055648, 2.52291330431898173709159151868, 3.32436871777128967980853293130, 3.90064479321008421013119269228, 4.98537747832813261903967458496, 5.96786533276493563587428428485, 7.15526235699589602664725007762, 7.48481692501882319981493975593, 8.475977037886668322684086559009
