| L(s) = 1 | + 2-s + 4-s − 2.85·5-s − 4.48·7-s + 8-s − 2.85·10-s − 0.198·11-s + 3.70·13-s − 4.48·14-s + 16-s + 0.618·17-s + 0.259·19-s − 2.85·20-s − 0.198·22-s + 3.16·25-s + 3.70·26-s − 4.48·28-s + 7.49·29-s − 2.80·31-s + 32-s + 0.618·34-s + 12.8·35-s − 5.90·37-s + 0.259·38-s − 2.85·40-s + 8.45·41-s + 10.0·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.27·5-s − 1.69·7-s + 0.353·8-s − 0.903·10-s − 0.0598·11-s + 1.02·13-s − 1.19·14-s + 0.250·16-s + 0.150·17-s + 0.0594·19-s − 0.638·20-s − 0.0422·22-s + 0.632·25-s + 0.727·26-s − 0.847·28-s + 1.39·29-s − 0.504·31-s + 0.176·32-s + 0.106·34-s + 2.16·35-s − 0.971·37-s + 0.0420·38-s − 0.451·40-s + 1.32·41-s + 1.53·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 + 4.48T + 7T^{2} \) |
| 11 | \( 1 + 0.198T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 - 0.259T + 19T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 - 8.45T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 9.81T + 47T^{2} \) |
| 53 | \( 1 - 3.93T + 53T^{2} \) |
| 59 | \( 1 + 9.81T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 - 7.35T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 9.62T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.16205823617621029581885665655, −6.64219428756007908194415426762, −6.05775447537360391963776349814, −5.33261962896839941841081568509, −4.25829148438884883327294451682, −3.84748037571141179920378009213, −3.21818616307498567465442340181, −2.61411464643538677586079494743, −1.10410397837064323384246963432, 0,
1.10410397837064323384246963432, 2.61411464643538677586079494743, 3.21818616307498567465442340181, 3.84748037571141179920378009213, 4.25829148438884883327294451682, 5.33261962896839941841081568509, 6.05775447537360391963776349814, 6.64219428756007908194415426762, 7.16205823617621029581885665655
