| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 4·11-s + 6·13-s + 16-s + 2·17-s − 20-s + 4·22-s + 25-s − 6·26-s − 6·29-s − 8·31-s − 32-s − 2·34-s − 10·37-s + 40-s + 2·41-s + 4·43-s − 4·44-s + 8·47-s − 50-s + 6·52-s + 2·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1.64·37-s + 0.158·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 1.16·47-s − 0.141·50-s + 0.832·52-s + 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.957237096013159459080835086230, −7.50793050530402086317500860857, −6.71654507983987973017676739321, −5.73468162934481500102414565629, −5.29992032847947121483959754241, −3.94449175269007332055216124286, −3.39915186610779589390347542849, −2.30609205239508517525684039531, −1.25635141165548889358757283011, 0,
1.25635141165548889358757283011, 2.30609205239508517525684039531, 3.39915186610779589390347542849, 3.94449175269007332055216124286, 5.29992032847947121483959754241, 5.73468162934481500102414565629, 6.71654507983987973017676739321, 7.50793050530402086317500860857, 7.957237096013159459080835086230
