| L(s) = 1 | − 2-s − 3-s + 4-s − 1.34·5-s + 6-s − 8-s + 9-s + 1.34·10-s + 3.25·11-s − 12-s − 3.34·13-s + 1.34·15-s + 16-s + 17-s − 18-s + 2.73·19-s − 1.34·20-s − 3.25·22-s + 1.32·23-s + 24-s − 3.17·25-s + 3.34·26-s − 27-s − 1.90·29-s − 1.34·30-s − 2.09·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.603·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.426·10-s + 0.982·11-s − 0.288·12-s − 0.929·13-s + 0.348·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.627·19-s − 0.301·20-s − 0.694·22-s + 0.275·23-s + 0.204·24-s − 0.635·25-s + 0.656·26-s − 0.192·27-s − 0.354·29-s − 0.246·30-s − 0.375·31-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
| \(L(1)\) |
\(\approx\) |
\(0.8386311577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8386311577\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 - 1.32T + 23T^{2} \) |
| 29 | \( 1 + 1.90T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 5.84T + 37T^{2} \) |
| 41 | \( 1 + 7.43T + 41T^{2} \) |
| 43 | \( 1 - 6.91T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 + 3.68T + 61T^{2} \) |
| 67 | \( 1 + 6.78T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 + 4.76T + 83T^{2} \) |
| 89 | \( 1 - 4.43T + 89T^{2} \) |
| 97 | \( 1 - 5.16T + 97T^{2} \) |
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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
−8.161101000660567584709717981962, −7.37387189347651467215671212773, −7.13156900611347046449326585591, −6.11562293277967033808567136650, −5.50091818742344070561212349531, −4.48965024087126798221992058552, −3.79284789426554225161974370036, −2.76580563325471965654633476100, −1.62529552843350456836571674612, −0.58922153763451302634658408400,
0.58922153763451302634658408400, 1.62529552843350456836571674612, 2.76580563325471965654633476100, 3.79284789426554225161974370036, 4.48965024087126798221992058552, 5.50091818742344070561212349531, 6.11562293277967033808567136650, 7.13156900611347046449326585591, 7.37387189347651467215671212773, 8.161101000660567584709717981962
