| L(s) = 1 | − 4·7-s − 11-s + 4·13-s + 4·19-s + 6·23-s + 6·29-s − 8·31-s − 2·37-s − 6·41-s + 8·43-s − 6·47-s + 9·49-s − 6·53-s − 12·59-s + 2·61-s − 10·67-s − 12·71-s + 16·73-s + 4·77-s − 8·79-s − 6·89-s − 16·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.301·11-s + 1.10·13-s + 0.917·19-s + 1.25·23-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.937·41-s + 1.21·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s − 1.22·67-s − 1.42·71-s + 1.87·73-s + 0.455·77-s − 0.900·79-s − 0.635·89-s − 1.67·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.33507922166391, −14.38344950679363, −14.01751687741906, −13.32218139088711, −13.06931025907777, −12.57390682121303, −12.01351272230430, −11.35619934048331, −10.75732307339310, −10.42059000527371, −9.637590406424339, −9.318825831085320, −8.765169905019123, −8.198794520401858, −7.308932762400042, −7.060691176228205, −6.242990124173315, −5.978424516020034, −5.208990561398557, −4.579930647393716, −3.682057629123003, −3.194036389682206, −2.866700115986540, −1.721157636824429, −0.9339088042695684, 0,
0.9339088042695684, 1.721157636824429, 2.866700115986540, 3.194036389682206, 3.682057629123003, 4.579930647393716, 5.208990561398557, 5.978424516020034, 6.242990124173315, 7.060691176228205, 7.308932762400042, 8.198794520401858, 8.765169905019123, 9.318825831085320, 9.637590406424339, 10.42059000527371, 10.75732307339310, 11.35619934048331, 12.01351272230430, 12.57390682121303, 13.06931025907777, 13.32218139088711, 14.01751687741906, 14.38344950679363, 15.33507922166391
