| L(s) = 1 | + 2·5-s − 4·7-s + 13-s − 2·17-s − 8·19-s + 8·23-s − 25-s + 2·29-s − 4·31-s − 8·35-s − 10·37-s − 2·41-s + 4·43-s − 12·47-s + 9·49-s − 6·53-s − 2·61-s + 2·65-s − 8·67-s − 12·71-s + 10·73-s + 8·79-s − 4·85-s + 14·89-s − 4·91-s − 16·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 0.277·13-s − 0.485·17-s − 1.83·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 1.35·35-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.256·61-s + 0.248·65-s − 0.977·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s − 0.433·85-s + 1.48·89-s − 0.419·91-s − 1.64·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.158792946221366931900105912370, −8.209770136822864596347152768350, −6.84717778862769781936733328882, −6.57432395350568485912302103534, −5.78674978184567296447754787702, −4.80441249928361298007080575834, −3.67113970451719789938663633179, −2.81347328384033747404310399701, −1.75350296463647814243579299746, 0,
1.75350296463647814243579299746, 2.81347328384033747404310399701, 3.67113970451719789938663633179, 4.80441249928361298007080575834, 5.78674978184567296447754787702, 6.57432395350568485912302103534, 6.84717778862769781936733328882, 8.209770136822864596347152768350, 9.158792946221366931900105912370
