L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 2·15-s + 2·17-s − 8·19-s + 4·21-s + 8·23-s − 25-s − 27-s − 2·29-s − 4·31-s − 8·35-s + 10·37-s − 2·41-s − 4·43-s + 2·45-s + 12·47-s + 9·49-s − 2·51-s + 6·53-s + 8·57-s − 2·61-s − 4·63-s − 8·67-s − 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.516·15-s + 0.485·17-s − 1.83·19-s + 0.872·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 1.35·35-s + 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 1.75·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 1.05·57-s − 0.256·61-s − 0.503·63-s − 0.977·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.270263691\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270263691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.695323695070305876621356025270, −7.44372178284179609930475636019, −6.83062460708306911889787953306, −6.05838628354437112387058390242, −5.81332067313138220793489914527, −4.74985294587632508321672237898, −3.83350425236925028468394065431, −2.90404888670441570594440478650, −2.00193178238602985630231809281, −0.64519645577139433460363939109,
0.64519645577139433460363939109, 2.00193178238602985630231809281, 2.90404888670441570594440478650, 3.83350425236925028468394065431, 4.74985294587632508321672237898, 5.81332067313138220793489914527, 6.05838628354437112387058390242, 6.83062460708306911889787953306, 7.44372178284179609930475636019, 8.695323695070305876621356025270