| L(s) = 1 | − 1.31·3-s − 3.46·7-s − 1.27·9-s − 11-s + 6.09·13-s − 3.46·17-s − 4·19-s + 4.54·21-s + 8.24·23-s + 5.61·27-s + 6.54·29-s − 1.72·31-s + 1.31·33-s + 8.24·37-s − 8·39-s − 6.54·41-s − 3.46·43-s + 2.62·47-s + 4.99·49-s + 4.54·51-s + 5.25·57-s − 14.2·59-s + 6.54·61-s + 4.41·63-s + 10.8·67-s − 10.8·69-s + 2.27·71-s + ⋯ |
| L(s) = 1 | − 0.758·3-s − 1.30·7-s − 0.424·9-s − 0.301·11-s + 1.68·13-s − 0.840·17-s − 0.917·19-s + 0.992·21-s + 1.71·23-s + 1.08·27-s + 1.21·29-s − 0.309·31-s + 0.228·33-s + 1.35·37-s − 1.28·39-s − 1.02·41-s − 0.528·43-s + 0.383·47-s + 0.714·49-s + 0.637·51-s + 0.695·57-s − 1.85·59-s + 0.838·61-s + 0.556·63-s + 1.32·67-s − 1.30·69-s + 0.269·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.31T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8.24T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 2.27T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 - 1.78T + 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−8.178422790193315275113476727568, −6.84757540744091159269381891012, −6.51072998240746023227036771926, −5.98263947862472572210560182945, −5.13275424558852444789102827200, −4.24623516449504814743372689102, −3.29682459697761996410726350899, −2.63829645003345684366048213591, −1.10969843869626480259056340581, 0,
1.10969843869626480259056340581, 2.63829645003345684366048213591, 3.29682459697761996410726350899, 4.24623516449504814743372689102, 5.13275424558852444789102827200, 5.98263947862472572210560182945, 6.51072998240746023227036771926, 6.84757540744091159269381891012, 8.178422790193315275113476727568
