L(s) = 1 | − 1.56e5·3-s − 1.90e8·5-s − 4.80e9·7-s − 6.97e10·9-s − 1.33e12·11-s − 9.05e12·13-s + 2.97e13·15-s + 4.05e13·17-s + 5.13e14·19-s + 7.50e14·21-s + 3.70e15·23-s + 2.44e16·25-s + 2.56e16·27-s − 5.34e15·29-s − 1.61e17·31-s + 2.08e17·33-s + 9.16e17·35-s − 9.84e17·37-s + 1.41e18·39-s + 3.74e18·41-s + 7.77e17·43-s + 1.32e19·45-s + 2.36e19·47-s − 4.26e18·49-s − 6.32e18·51-s − 5.09e19·53-s + 2.54e20·55-s + ⋯ |
L(s) = 1 | − 0.509·3-s − 1.74·5-s − 0.918·7-s − 0.740·9-s − 1.40·11-s − 1.40·13-s + 0.888·15-s + 0.286·17-s + 1.01·19-s + 0.467·21-s + 0.810·23-s + 2.04·25-s + 0.886·27-s − 0.0812·29-s − 1.14·31-s + 0.716·33-s + 1.60·35-s − 0.909·37-s + 0.713·39-s + 1.06·41-s + 0.127·43-s + 1.29·45-s + 1.39·47-s − 0.155·49-s − 0.145·51-s − 0.755·53-s + 2.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(12)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{25}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 1.56e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.90e8T + 1.19e16T^{2} \) |
| 7 | \( 1 + 4.80e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.33e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 9.05e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 4.05e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 5.13e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 3.70e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 5.34e15T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.61e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 9.84e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 3.74e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 7.77e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.36e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 5.09e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 3.99e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 4.81e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.52e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 1.07e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 1.32e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.09e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.68e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 1.71e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 7.38e22T + 4.96e45T^{2} \) |
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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
−10.34012632091204019969653495423, −8.989920236553066610931570248413, −7.68207874934686475857120383996, −7.19755416615285294748383182697, −5.59028978554926910398488640525, −4.73714799434028549015697012888, −3.35442163735016874068001227676, −2.72470381915829223371576169257, −0.57799906427386718623503042335, 0,
0.57799906427386718623503042335, 2.72470381915829223371576169257, 3.35442163735016874068001227676, 4.73714799434028549015697012888, 5.59028978554926910398488640525, 7.19755416615285294748383182697, 7.68207874934686475857120383996, 8.989920236553066610931570248413, 10.34012632091204019969653495423