| L(s) = 1 | − 3-s − 4.17·5-s − 2.39·7-s + 9-s − 0.984·11-s − 5.39·13-s + 4.17·15-s − 1.54·17-s + 1.71·19-s + 2.39·21-s + 5.82·23-s + 12.4·25-s − 27-s + 7.84·29-s − 5.59·31-s + 0.984·33-s + 9.99·35-s + 11.3·37-s + 5.39·39-s − 6.86·41-s + 3.55·43-s − 4.17·45-s + 4.90·47-s − 1.26·49-s + 1.54·51-s + 1.43·53-s + 4.11·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.86·5-s − 0.905·7-s + 0.333·9-s − 0.296·11-s − 1.49·13-s + 1.07·15-s − 0.374·17-s + 0.393·19-s + 0.522·21-s + 1.21·23-s + 2.48·25-s − 0.192·27-s + 1.45·29-s − 1.00·31-s + 0.171·33-s + 1.69·35-s + 1.86·37-s + 0.863·39-s − 1.07·41-s + 0.542·43-s − 0.622·45-s + 0.714·47-s − 0.180·49-s + 0.216·51-s + 0.197·53-s + 0.554·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 11 | \( 1 + 0.984T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 1.54T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 - 7.84T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 - 2.04T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 - 0.582T + 67T^{2} \) |
| 71 | \( 1 + 4.37T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 - 7.66T + 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.61498690922099493679550918360, −7.02235040822238422909404442615, −6.59573208274118334889435521237, −5.41830776472529893136782069197, −4.72610117002063664981113419633, −4.17281767453836576857763309551, −3.21829015057200278348922337639, −2.63718081543174518697832952075, −0.827633521603719245143926553171, 0,
0.827633521603719245143926553171, 2.63718081543174518697832952075, 3.21829015057200278348922337639, 4.17281767453836576857763309551, 4.72610117002063664981113419633, 5.41830776472529893136782069197, 6.59573208274118334889435521237, 7.02235040822238422909404442615, 7.61498690922099493679550918360
