| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 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
−7.55584980203469143488929622574, −6.73116486268380092990656122077, −6.35214200574868265263274577271, −5.47025294806105510476455863715, −4.79032797746265243503127012091, −4.09981457995011309139565657186, −3.17499886546837037551078882308, −2.25981540524048606317574899903, −1.17984480931399129360135920265, 0,
1.17984480931399129360135920265, 2.25981540524048606317574899903, 3.17499886546837037551078882308, 4.09981457995011309139565657186, 4.79032797746265243503127012091, 5.47025294806105510476455863715, 6.35214200574868265263274577271, 6.73116486268380092990656122077, 7.55584980203469143488929622574
