| L(s) = 1 | − 3-s − 3.18·5-s − 1.57·7-s + 9-s + 3.18·11-s + 4.33·13-s + 3.18·15-s − 4.15·17-s − 2.44·19-s + 1.57·21-s + 7.36·23-s + 5.15·25-s − 27-s + 6.11·29-s − 8.91·31-s − 3.18·33-s + 5.00·35-s − 10.0·37-s − 4.33·39-s + 10.5·41-s + 5.14·43-s − 3.18·45-s + 1.65·47-s − 4.53·49-s + 4.15·51-s − 5.04·53-s − 10.1·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.42·5-s − 0.593·7-s + 0.333·9-s + 0.960·11-s + 1.20·13-s + 0.822·15-s − 1.00·17-s − 0.560·19-s + 0.342·21-s + 1.53·23-s + 1.03·25-s − 0.192·27-s + 1.13·29-s − 1.60·31-s − 0.554·33-s + 0.846·35-s − 1.65·37-s − 0.694·39-s + 1.65·41-s + 0.784·43-s − 0.475·45-s + 0.241·47-s − 0.647·49-s + 0.581·51-s − 0.692·53-s − 1.36·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 - 6.11T + 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 5.04T + 53T^{2} \) |
| 59 | \( 1 + 3.76T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 + 1.31T + 67T^{2} \) |
| 71 | \( 1 - 1.93T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 7.44T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 7.00T + 97T^{2} \) |
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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
−8.873490215998430461073762884670, −7.992493686149664903660593881318, −6.92243848735792611064607713220, −6.66540325331779683988239516169, −5.62831285336256198365195421758, −4.43177027449339330481338052985, −3.93616874991685186975812144854, −3.04905140019430320843415677236, −1.30352337112351417373045090810, 0,
1.30352337112351417373045090810, 3.04905140019430320843415677236, 3.93616874991685186975812144854, 4.43177027449339330481338052985, 5.62831285336256198365195421758, 6.66540325331779683988239516169, 6.92243848735792611064607713220, 7.992493686149664903660593881318, 8.873490215998430461073762884670
