| L(s) = 1 | − 2-s + 3-s + 4-s + 0.414·5-s − 6-s − 8-s + 9-s − 0.414·10-s − 0.828·11-s + 12-s − 5.65·13-s + 0.414·15-s + 16-s − 17-s − 18-s + 3·19-s + 0.414·20-s + 0.828·22-s + 4.41·23-s − 24-s − 4.82·25-s + 5.65·26-s + 27-s − 6.82·29-s − 0.414·30-s + 6·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.185·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.130·10-s − 0.249·11-s + 0.288·12-s − 1.56·13-s + 0.106·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.688·19-s + 0.0926·20-s + 0.176·22-s + 0.920·23-s − 0.204·24-s − 0.965·25-s + 1.10·26-s + 0.192·27-s − 1.26·29-s − 0.0756·30-s + 1.07·31-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
| \(L(1)\) |
\(\approx\) |
\(1.537502488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537502488\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.414T + 5T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 9.82T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 0.656T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 - 4.65T + 67T^{2} \) |
| 71 | \( 1 - 2.41T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 + 9.65T + 97T^{2} \) |
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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
−8.188088343202154938095708951230, −7.47587109513816898410067721729, −7.24327488072229659884759586446, −6.19236398308072811211386099558, −5.35208339480885541967817547688, −4.57161425307637155050470365437, −3.52598226732704934823301771248, −2.59306511475149919301252106104, −2.04160761143890824091897645982, −0.71415710724654164417893830589,
0.71415710724654164417893830589, 2.04160761143890824091897645982, 2.59306511475149919301252106104, 3.52598226732704934823301771248, 4.57161425307637155050470365437, 5.35208339480885541967817547688, 6.19236398308072811211386099558, 7.24327488072229659884759586446, 7.47587109513816898410067721729, 8.188088343202154938095708951230
