| L(s) = 1 | + 22·2-s − 27·3-s + 356·4-s − 594·6-s + 420·7-s + 5.01e3·8-s + 729·9-s − 2.94e3·11-s − 9.61e3·12-s + 1.10e4·13-s + 9.24e3·14-s + 6.47e4·16-s + 1.65e4·17-s + 1.60e4·18-s − 2.53e4·19-s − 1.13e4·21-s − 6.47e4·22-s + 5.88e3·23-s − 1.35e5·24-s + 2.42e5·26-s − 1.96e4·27-s + 1.49e5·28-s + 1.63e5·29-s − 2.01e5·31-s + 7.83e5·32-s + 7.94e4·33-s + 3.64e5·34-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.577·3-s + 2.78·4-s − 1.12·6-s + 0.462·7-s + 3.46·8-s + 1/3·9-s − 0.666·11-s − 1.60·12-s + 1.38·13-s + 0.899·14-s + 3.95·16-s + 0.816·17-s + 0.648·18-s − 0.848·19-s − 0.267·21-s − 1.29·22-s + 0.100·23-s − 1.99·24-s + 2.70·26-s − 0.192·27-s + 1.28·28-s + 1.24·29-s − 1.21·31-s + 4.22·32-s + 0.385·33-s + 1.58·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.058141173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.058141173\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 11 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 60 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 2944 T + p^{7} T^{2} \) |
| 13 | \( 1 - 11006 T + p^{7} T^{2} \) |
| 17 | \( 1 - 16546 T + p^{7} T^{2} \) |
| 19 | \( 1 + 25364 T + p^{7} T^{2} \) |
| 23 | \( 1 - 5880 T + p^{7} T^{2} \) |
| 29 | \( 1 - 163042 T + p^{7} T^{2} \) |
| 31 | \( 1 + 201600 T + p^{7} T^{2} \) |
| 37 | \( 1 + 120530 T + p^{7} T^{2} \) |
| 41 | \( 1 + 115910 T + p^{7} T^{2} \) |
| 43 | \( 1 - 19148 T + p^{7} T^{2} \) |
| 47 | \( 1 + 841016 T + p^{7} T^{2} \) |
| 53 | \( 1 + 501890 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1586176 T + p^{7} T^{2} \) |
| 61 | \( 1 + 372962 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4561044 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1512832 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1522910 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4231920 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1854204 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6888174 T + p^{7} T^{2} \) |
| 97 | \( 1 + 3700034 T + p^{7} T^{2} \) |
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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
−13.09838024159491212740692389062, −12.26224215684148444250257101961, −11.18816141162080707728413033140, −10.52798791474975947998997701691, −7.994038182769346481815521741366, −6.59431903875824013712814002075, −5.63306850433352736404406217542, −4.59743156890007725735576346748, −3.29866953313571255979108238926, −1.60130663021197977401507554934,
1.60130663021197977401507554934, 3.29866953313571255979108238926, 4.59743156890007725735576346748, 5.63306850433352736404406217542, 6.59431903875824013712814002075, 7.994038182769346481815521741366, 10.52798791474975947998997701691, 11.18816141162080707728413033140, 12.26224215684148444250257101961, 13.09838024159491212740692389062
