L(s) = 1 | + 2·5-s − 7-s + 4·11-s + 2·13-s + 6·17-s + 19-s − 25-s + 10·29-s + 8·31-s − 2·35-s − 2·37-s − 6·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s + 8·55-s + 4·59-s − 10·61-s + 4·65-s + 8·67-s − 12·71-s − 14·73-s − 4·77-s − 4·79-s + 4·83-s + 12·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.229·19-s − 1/5·25-s + 1.85·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 0.520·59-s − 1.28·61-s + 0.496·65-s + 0.977·67-s − 1.42·71-s − 1.63·73-s − 0.455·77-s − 0.450·79-s + 0.439·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.178757548\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.178757548\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.65478047116696422669303448072, −6.87000862610472789045071680173, −6.16482505023945957063098815526, −5.91960408483394904514317921926, −4.96762823348528019770922126026, −4.18290305064466179578328771321, −3.34584100967352602302155182341, −2.70134925506296298553437922558, −1.53860957248944911385408617949, −0.953031729197470419184280734853,
0.953031729197470419184280734853, 1.53860957248944911385408617949, 2.70134925506296298553437922558, 3.34584100967352602302155182341, 4.18290305064466179578328771321, 4.96762823348528019770922126026, 5.91960408483394904514317921926, 6.16482505023945957063098815526, 6.87000862610472789045071680173, 7.65478047116696422669303448072