| L(s) = 1 | + 4.48·2-s − 8.90·3-s + 12.1·4-s + 2.08·5-s − 39.9·6-s + 18.5·8-s + 52.2·9-s + 9.35·10-s + 33.0·11-s − 108.·12-s − 89.3·13-s − 18.5·15-s − 13.7·16-s + 46.0·17-s + 234.·18-s + 89.4·19-s + 25.3·20-s + 148.·22-s + 23·23-s − 165.·24-s − 120.·25-s − 400.·26-s − 224.·27-s + 152.·29-s − 83.2·30-s − 165.·31-s − 210.·32-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 1.71·3-s + 1.51·4-s + 0.186·5-s − 2.71·6-s + 0.821·8-s + 1.93·9-s + 0.295·10-s + 0.905·11-s − 2.59·12-s − 1.90·13-s − 0.319·15-s − 0.214·16-s + 0.656·17-s + 3.07·18-s + 1.07·19-s + 0.282·20-s + 1.43·22-s + 0.208·23-s − 1.40·24-s − 0.965·25-s − 3.02·26-s − 1.60·27-s + 0.975·29-s − 0.506·30-s − 0.958·31-s − 1.16·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 4.48T + 8T^{2} \) |
| 3 | \( 1 + 8.90T + 27T^{2} \) |
| 5 | \( 1 - 2.08T + 125T^{2} \) |
| 11 | \( 1 - 33.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 89.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 89.4T + 6.85e3T^{2} \) |
| 29 | \( 1 - 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 237.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 246.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 453.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 475.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 149.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 126.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 468.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 34.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 198.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 301.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 719.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 828.T + 9.12e5T^{2} \) |
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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
−9.530006664398336846464833926365, −7.63222197630273622391786770956, −6.89519208970483892682154644579, −6.19786064781811315987967293054, −5.42061182866943681009433943219, −4.91560473178082067730526317726, −4.14480567803632146430273247358, −2.92748782660858983786365851181, −1.48913854967644232817860899623, 0,
1.48913854967644232817860899623, 2.92748782660858983786365851181, 4.14480567803632146430273247358, 4.91560473178082067730526317726, 5.42061182866943681009433943219, 6.19786064781811315987967293054, 6.89519208970483892682154644579, 7.63222197630273622391786770956, 9.530006664398336846464833926365
