| L(s) = 1 | − 5-s − 7-s + 2·13-s − 2·19-s + 25-s + 6·29-s − 8·31-s + 35-s − 4·37-s + 6·41-s − 2·43-s + 6·47-s + 49-s + 6·53-s − 12·59-s + 8·61-s − 2·65-s − 2·67-s − 6·71-s + 2·73-s + 16·79-s + 6·89-s − 2·91-s + 2·95-s − 10·97-s + 6·101-s − 8·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.554·13-s − 0.458·19-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s − 0.657·37-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 1.02·61-s − 0.248·65-s − 0.244·67-s − 0.712·71-s + 0.234·73-s + 1.80·79-s + 0.635·89-s − 0.209·91-s + 0.205·95-s − 1.01·97-s + 0.597·101-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.490891550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490891550\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.270207471199591506248398971090, −7.49756017971268270939194991270, −6.82891222608894401755323656543, −6.11415456732748664188676934089, −5.36462460741365631953110507100, −4.42948391079815388966193914877, −3.74129993568519053731492381467, −2.96142925173523128868445834982, −1.90860795407291261876798047655, −0.66084113438639072196420286884,
0.66084113438639072196420286884, 1.90860795407291261876798047655, 2.96142925173523128868445834982, 3.74129993568519053731492381467, 4.42948391079815388966193914877, 5.36462460741365631953110507100, 6.11415456732748664188676934089, 6.82891222608894401755323656543, 7.49756017971268270939194991270, 8.270207471199591506248398971090
