L(s) = 1 | − 1.49·3-s + 1.26·5-s − 0.770·9-s + 11-s − 6.62·13-s − 1.88·15-s + 1.02·17-s − 2.10·19-s + 1.84·23-s − 3.40·25-s + 5.63·27-s + 4.86·29-s + 1.60·31-s − 1.49·33-s − 2.16·37-s + 9.88·39-s + 6.49·41-s + 12.3·43-s − 0.974·45-s + 5.02·47-s − 1.53·51-s + 2.52·53-s + 1.26·55-s + 3.13·57-s − 3.54·59-s + 2.94·61-s − 8.36·65-s + ⋯ |
L(s) = 1 | − 0.861·3-s + 0.565·5-s − 0.256·9-s + 0.301·11-s − 1.83·13-s − 0.487·15-s + 0.248·17-s − 0.482·19-s + 0.385·23-s − 0.680·25-s + 1.08·27-s + 0.903·29-s + 0.289·31-s − 0.259·33-s − 0.355·37-s + 1.58·39-s + 1.01·41-s + 1.88·43-s − 0.145·45-s + 0.733·47-s − 0.214·51-s + 0.346·53-s + 0.170·55-s + 0.415·57-s − 0.460·59-s + 0.377·61-s − 1.03·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 - 1.26T + 5T^{2} \) |
| 13 | \( 1 + 6.62T + 13T^{2} \) |
| 17 | \( 1 - 1.02T + 17T^{2} \) |
| 19 | \( 1 + 2.10T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 - 4.86T + 29T^{2} \) |
| 31 | \( 1 - 1.60T + 31T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 5.02T + 47T^{2} \) |
| 53 | \( 1 - 2.52T + 53T^{2} \) |
| 59 | \( 1 + 3.54T + 59T^{2} \) |
| 61 | \( 1 - 2.94T + 61T^{2} \) |
| 67 | \( 1 - 9.84T + 67T^{2} \) |
| 71 | \( 1 + 8.86T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 8.31T + 79T^{2} \) |
| 83 | \( 1 + 4.54T + 83T^{2} \) |
| 89 | \( 1 + 6.62T + 89T^{2} \) |
| 97 | \( 1 + 1.10T + 97T^{2} \) |
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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
−7.29070410930602769555610865643, −6.69065713629950602204171383386, −5.90839471544056526543034140835, −5.49653793262762810085980324564, −4.73591945731191907365039470785, −4.11772178022066034019138018331, −2.79781902170812333856919235267, −2.34222466721119668011254353231, −1.07893465019443229830299615489, 0,
1.07893465019443229830299615489, 2.34222466721119668011254353231, 2.79781902170812333856919235267, 4.11772178022066034019138018331, 4.73591945731191907365039470785, 5.49653793262762810085980324564, 5.90839471544056526543034140835, 6.69065713629950602204171383386, 7.29070410930602769555610865643