| L(s) = 1 | − 5-s − 3·11-s − 13-s − 3·17-s + 8·19-s + 7·23-s + 25-s − 6·29-s + 2·31-s + 37-s − 5·41-s + 10·43-s − 6·47-s + 6·53-s + 3·55-s − 15·59-s − 2·61-s + 65-s + 3·67-s − 73-s + 9·79-s + 6·83-s + 3·85-s − 3·89-s − 8·95-s + 9·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.904·11-s − 0.277·13-s − 0.727·17-s + 1.83·19-s + 1.45·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.164·37-s − 0.780·41-s + 1.52·43-s − 0.875·47-s + 0.824·53-s + 0.404·55-s − 1.95·59-s − 0.256·61-s + 0.124·65-s + 0.366·67-s − 0.117·73-s + 1.01·79-s + 0.658·83-s + 0.325·85-s − 0.317·89-s − 0.820·95-s + 0.913·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.21203041450429, −12.66639827960379, −12.29275385884840, −11.70709736944525, −11.27368713282442, −10.94438068623500, −10.44903714252693, −9.879323787935184, −9.336163895167429, −9.060519955553848, −8.472841947454963, −7.790494201396245, −7.519129645831042, −7.175489438756602, −6.531760064834850, −5.937088886786093, −5.327510503258317, −4.948148384540940, −4.568118462436455, −3.734175901843141, −3.286171134983734, −2.775107201856947, −2.229064657683114, −1.391740206617801, −0.7566023944783959, 0,
0.7566023944783959, 1.391740206617801, 2.229064657683114, 2.775107201856947, 3.286171134983734, 3.734175901843141, 4.568118462436455, 4.948148384540940, 5.327510503258317, 5.937088886786093, 6.531760064834850, 7.175489438756602, 7.519129645831042, 7.790494201396245, 8.472841947454963, 9.060519955553848, 9.336163895167429, 9.879323787935184, 10.44903714252693, 10.94438068623500, 11.27368713282442, 11.70709736944525, 12.29275385884840, 12.66639827960379, 13.21203041450429
