| L(s) = 1 | + 3-s + 3.46·5-s − 7-s + 9-s − 1.46·11-s + 2·13-s + 3.46·15-s + 0.535·17-s + 6.92·19-s − 21-s − 1.46·23-s + 6.99·25-s + 27-s − 4.92·29-s − 10.9·31-s − 1.46·33-s − 3.46·35-s − 2·37-s + 2·39-s + 11.4·41-s − 8·43-s + 3.46·45-s + 10.9·47-s + 49-s + 0.535·51-s − 2·53-s − 5.07·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.54·5-s − 0.377·7-s + 0.333·9-s − 0.441·11-s + 0.554·13-s + 0.894·15-s + 0.129·17-s + 1.58·19-s − 0.218·21-s − 0.305·23-s + 1.39·25-s + 0.192·27-s − 0.915·29-s − 1.96·31-s − 0.254·33-s − 0.585·35-s − 0.328·37-s + 0.320·39-s + 1.79·41-s − 1.21·43-s + 0.516·45-s + 1.59·47-s + 0.142·49-s + 0.0750·51-s − 0.274·53-s − 0.683·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.291605718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.291605718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 1.07T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−10.33766677607814470408512191775, −9.379615572406319797328090655477, −9.225638526702577970973858034414, −7.87523126180874102016645820385, −7.00413869600387648617253226188, −5.84833168662077872258392685058, −5.33227847595585265381325435022, −3.73204424624066476989619424617, −2.65386364635551658183695782130, −1.53667169342930680116766233112,
1.53667169342930680116766233112, 2.65386364635551658183695782130, 3.73204424624066476989619424617, 5.33227847595585265381325435022, 5.84833168662077872258392685058, 7.00413869600387648617253226188, 7.87523126180874102016645820385, 9.225638526702577970973858034414, 9.379615572406319797328090655477, 10.33766677607814470408512191775
