| L(s) = 1 | − 3·9-s − 6·13-s − 8·17-s − 4·29-s − 2·37-s + 10·41-s − 7·49-s + 14·53-s + 12·61-s + 16·73-s + 9·81-s − 10·89-s + 8·97-s + 20·101-s + 20·109-s − 16·113-s + 18·117-s + ⋯ |
| L(s) = 1 | − 9-s − 1.66·13-s − 1.94·17-s − 0.742·29-s − 0.328·37-s + 1.56·41-s − 49-s + 1.92·53-s + 1.53·61-s + 1.87·73-s + 81-s − 1.05·89-s + 0.812·97-s + 1.99·101-s + 1.91·109-s − 1.50·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9325501921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9325501921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.035847445850808480947150543532, −7.26087671973770381388650597430, −6.70690390605105823264694056520, −5.87032195325798923409553352023, −5.13752015034940758406923559560, −4.50763465274742950079817013363, −3.62307452930435219633420944707, −2.47842716075231651048188342932, −2.20724184190000840116811085352, −0.46483908406341572736526436318,
0.46483908406341572736526436318, 2.20724184190000840116811085352, 2.47842716075231651048188342932, 3.62307452930435219633420944707, 4.50763465274742950079817013363, 5.13752015034940758406923559560, 5.87032195325798923409553352023, 6.70690390605105823264694056520, 7.26087671973770381388650597430, 8.035847445850808480947150543532
