| L(s) = 1 | − 1.82·2-s + 1.95·3-s + 2.33·4-s − 3.57·6-s + 0.209·7-s − 2.44·8-s + 2.82·9-s + 4.57·12-s − 0.381·14-s + 2.12·16-s − 1.33·17-s − 5.16·18-s + 0.408·21-s − 4.78·24-s + 3.57·27-s + 0.488·28-s − 1.44·32-s + 2.44·34-s + 6.61·36-s − 0.618·37-s − 0.747·42-s − 47-s + 4.16·48-s − 0.956·49-s − 2.61·51-s + 1.61·53-s − 6.53·54-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 1.95·3-s + 2.33·4-s − 3.57·6-s + 0.209·7-s − 2.44·8-s + 2.82·9-s + 4.57·12-s − 0.381·14-s + 2.12·16-s − 1.33·17-s − 5.16·18-s + 0.408·21-s − 4.78·24-s + 3.57·27-s + 0.488·28-s − 1.44·32-s + 2.44·34-s + 6.61·36-s − 0.618·37-s − 0.747·42-s − 47-s + 4.16·48-s − 0.956·49-s − 2.61·51-s + 1.61·53-s − 6.53·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9370468385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9370468385\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + T^{2} \) |
| 3 | \( 1 - 1.95T + T^{2} \) |
| 7 | \( 1 - 0.209T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.33T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.618T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.61T + T^{2} \) |
| 59 | \( 1 + 0.209T + T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.95T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 - 1.61T + T^{2} \) |
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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
−9.684096355257312150756606192427, −8.891756573158130154929651242698, −8.639142445668157766414201289499, −7.81359251623725718508876178747, −7.21604439044104862806315225735, −6.45147765031395573509791799386, −4.52013621995076137473722639663, −3.27318480474347868327140193785, −2.34269363109503255909060822352, −1.57324829047597730314027601403,
1.57324829047597730314027601403, 2.34269363109503255909060822352, 3.27318480474347868327140193785, 4.52013621995076137473722639663, 6.45147765031395573509791799386, 7.21604439044104862806315225735, 7.81359251623725718508876178747, 8.639142445668157766414201289499, 8.891756573158130154929651242698, 9.684096355257312150756606192427
