L(s) = 1 | + 2-s − 2.35·3-s + 4-s − 2.69·5-s − 2.35·6-s + 2.49·7-s + 8-s + 2.55·9-s − 2.69·10-s + 3.60·11-s − 2.35·12-s + 2.49·14-s + 6.34·15-s + 16-s − 4.54·17-s + 2.55·18-s − 19-s − 2.69·20-s − 5.87·21-s + 3.60·22-s − 2.21·23-s − 2.35·24-s + 2.24·25-s + 1.04·27-s + 2.49·28-s − 6.98·29-s + 6.34·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.36·3-s + 0.5·4-s − 1.20·5-s − 0.962·6-s + 0.942·7-s + 0.353·8-s + 0.851·9-s − 0.851·10-s + 1.08·11-s − 0.680·12-s + 0.666·14-s + 1.63·15-s + 0.250·16-s − 1.10·17-s + 0.602·18-s − 0.229·19-s − 0.601·20-s − 1.28·21-s + 0.768·22-s − 0.462·23-s − 0.481·24-s + 0.449·25-s + 0.201·27-s + 0.471·28-s − 1.29·29-s + 1.15·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 23 | \( 1 + 2.21T + 23T^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 - 6.45T + 31T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 - 9.70T + 41T^{2} \) |
| 43 | \( 1 + 4.09T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 0.176T + 53T^{2} \) |
| 59 | \( 1 - 7.28T + 59T^{2} \) |
| 61 | \( 1 - 8.51T + 61T^{2} \) |
| 67 | \( 1 - 8.05T + 67T^{2} \) |
| 71 | \( 1 - 1.28T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 6.89T + 97T^{2} \) |
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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
−7.44129036339062049387721576513, −6.78709018872131247320173989319, −6.24804141632679922392897018103, −5.41778827928363124605856637405, −4.75603692531440310338277419968, −4.17073400233916579383744353451, −3.65968398085277745595043428312, −2.24626550932553019899619661285, −1.19109731128940926682999873647, 0,
1.19109731128940926682999873647, 2.24626550932553019899619661285, 3.65968398085277745595043428312, 4.17073400233916579383744353451, 4.75603692531440310338277419968, 5.41778827928363124605856637405, 6.24804141632679922392897018103, 6.78709018872131247320173989319, 7.44129036339062049387721576513