| L(s) = 1 | + 3-s + 0.697·7-s + 9-s + 0.697·11-s + 0.697·13-s + 5.60·17-s + 4.60·19-s + 0.697·21-s + 23-s + 27-s + 4.60·29-s − 1.90·31-s + 0.697·33-s − 2.39·37-s + 0.697·39-s + 3·41-s + 8.21·43-s − 12.8·47-s − 6.51·49-s + 5.60·51-s + 6.69·53-s + 4.60·57-s − 11.1·59-s + 3.90·61-s + 0.697·63-s + 10.9·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.263·7-s + 0.333·9-s + 0.210·11-s + 0.193·13-s + 1.35·17-s + 1.05·19-s + 0.152·21-s + 0.208·23-s + 0.192·27-s + 0.855·29-s − 0.342·31-s + 0.121·33-s − 0.393·37-s + 0.111·39-s + 0.468·41-s + 1.25·43-s − 1.86·47-s − 0.930·49-s + 0.784·51-s + 0.919·53-s + 0.610·57-s − 1.44·59-s + 0.500·61-s + 0.0878·63-s + 1.33·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.076333206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.076333206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 0.697T + 7T^{2} \) |
| 11 | \( 1 - 0.697T + 11T^{2} \) |
| 13 | \( 1 - 0.697T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 8.21T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 6.69T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 7.90T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 - 8.51T + 97T^{2} \) |
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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
−7.87357488533897013002727390135, −7.46528635607959023574881803384, −6.64862520402777687386837632388, −5.79966367448247100614622381844, −5.12301212585931815765441120316, −4.32254192684725966005979377820, −3.40291345847474480221791976269, −2.92205512450502461339488737367, −1.74772185666802049379711348219, −0.932539410280611531101875183833,
0.932539410280611531101875183833, 1.74772185666802049379711348219, 2.92205512450502461339488737367, 3.40291345847474480221791976269, 4.32254192684725966005979377820, 5.12301212585931815765441120316, 5.79966367448247100614622381844, 6.64862520402777687386837632388, 7.46528635607959023574881803384, 7.87357488533897013002727390135
