| L(s) = 1 | − 2-s + 4-s − 1.73·7-s − 8-s + 1.73·11-s + 4·13-s + 1.73·14-s + 16-s − 3.46·17-s − 1.73·22-s − 5·25-s − 4·26-s − 1.73·28-s − 3·29-s + 7·31-s − 32-s + 3.46·34-s − 12·41-s + 10.3·43-s + 1.73·44-s − 4·49-s + 5·50-s + 4·52-s + 8.66·53-s + 1.73·56-s + 3·58-s − 9·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.654·7-s − 0.353·8-s + 0.522·11-s + 1.10·13-s + 0.462·14-s + 0.250·16-s − 0.840·17-s − 0.369·22-s − 25-s − 0.784·26-s − 0.327·28-s − 0.557·29-s + 1.25·31-s − 0.176·32-s + 0.594·34-s − 1.87·41-s + 1.58·43-s + 0.261·44-s − 0.571·49-s + 0.707·50-s + 0.554·52-s + 1.18·53-s + 0.231·56-s + 0.393·58-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 - 6.92T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.32659997443610031800001009612, −6.68223561950816488926636694563, −6.18102937958026764801176895729, −5.55735975295730741062833071010, −4.40006159029003920580563387129, −3.75912150230060227923366018572, −2.97414438902044704698805205137, −2.03174953312738563647463918200, −1.14571956961391723653418744583, 0,
1.14571956961391723653418744583, 2.03174953312738563647463918200, 2.97414438902044704698805205137, 3.75912150230060227923366018572, 4.40006159029003920580563387129, 5.55735975295730741062833071010, 6.18102937958026764801176895729, 6.68223561950816488926636694563, 7.32659997443610031800001009612
