L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s + 2·13-s + 4·14-s + 16-s + 2·17-s − 19-s + 20-s + 25-s + 2·26-s + 4·28-s − 10·29-s + 32-s + 2·34-s + 4·35-s + 2·37-s − 38-s + 40-s − 2·41-s − 4·43-s + 9·49-s + 50-s + 2·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 1.85·29-s + 0.176·32-s + 0.342·34-s + 0.676·35-s + 0.328·37-s − 0.162·38-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + 9/7·49-s + 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.569184906\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.569184906\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.305177297450511271256974438638, −8.396145967730844971599373741212, −7.74299779225065503350659037807, −6.89121162523429679892474931966, −5.82351831026843318247931017497, −5.29384405955592108139129850733, −4.42243756533182632475893532486, −3.52537208444234564183609012151, −2.21785456744133671051657891609, −1.38919645818876595670946292707,
1.38919645818876595670946292707, 2.21785456744133671051657891609, 3.52537208444234564183609012151, 4.42243756533182632475893532486, 5.29384405955592108139129850733, 5.82351831026843318247931017497, 6.89121162523429679892474931966, 7.74299779225065503350659037807, 8.396145967730844971599373741212, 9.305177297450511271256974438638