| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 1.58·11-s − 12-s + 1.82·13-s − 15-s + 16-s + 17-s + 18-s − 2.24·19-s + 20-s − 1.58·22-s − 8.82·23-s − 24-s − 4·25-s + 1.82·26-s − 27-s − 5.41·29-s − 30-s − 0.242·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.478·11-s − 0.288·12-s + 0.507·13-s − 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.514·19-s + 0.223·20-s − 0.338·22-s − 1.84·23-s − 0.204·24-s − 0.800·25-s + 0.358·26-s − 0.192·27-s − 1.00·29-s − 0.182·30-s − 0.0435·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 0.242T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 4.89T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 3.58T + 79T^{2} \) |
| 83 | \( 1 + 5.58T + 83T^{2} \) |
| 89 | \( 1 + 5.34T + 89T^{2} \) |
| 97 | \( 1 + 5.82T + 97T^{2} \) |
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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
−7.900265414227790271720039308260, −6.86230406707792879823196882531, −6.34056837063142664947094121074, −5.55216263015654737539146589480, −5.20822995222430638991053522901, −4.07195053133499393012173342885, −3.58707330245954172800683386957, −2.29603981038630018112668121140, −1.62554399514333049626924143747, 0,
1.62554399514333049626924143747, 2.29603981038630018112668121140, 3.58707330245954172800683386957, 4.07195053133499393012173342885, 5.20822995222430638991053522901, 5.55216263015654737539146589480, 6.34056837063142664947094121074, 6.86230406707792879823196882531, 7.900265414227790271720039308260
