| L(s) = 1 | + 2-s + 0.216·3-s + 4-s + 0.737·5-s + 0.216·6-s + 0.342·7-s + 8-s − 2.95·9-s + 0.737·10-s − 11-s + 0.216·12-s − 1.84·13-s + 0.342·14-s + 0.159·15-s + 16-s − 7.54·17-s − 2.95·18-s + 6.47·19-s + 0.737·20-s + 0.0739·21-s − 22-s + 0.711·23-s + 0.216·24-s − 4.45·25-s − 1.84·26-s − 1.28·27-s + 0.342·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.124·3-s + 0.5·4-s + 0.329·5-s + 0.0882·6-s + 0.129·7-s + 0.353·8-s − 0.984·9-s + 0.233·10-s − 0.301·11-s + 0.0624·12-s − 0.511·13-s + 0.0914·14-s + 0.0411·15-s + 0.250·16-s − 1.82·17-s − 0.696·18-s + 1.48·19-s + 0.164·20-s + 0.0161·21-s − 0.213·22-s + 0.148·23-s + 0.0441·24-s − 0.891·25-s − 0.361·26-s − 0.247·27-s + 0.0646·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.216T + 3T^{2} \) |
| 5 | \( 1 - 0.737T + 5T^{2} \) |
| 7 | \( 1 - 0.342T + 7T^{2} \) |
| 13 | \( 1 + 1.84T + 13T^{2} \) |
| 17 | \( 1 + 7.54T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 0.711T + 23T^{2} \) |
| 29 | \( 1 + 3.71T + 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 7.49T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 5.38T + 53T^{2} \) |
| 59 | \( 1 - 5.04T + 59T^{2} \) |
| 61 | \( 1 + 9.27T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 6.11T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−7.80789867841411201324819395898, −7.30130387924704141598702750688, −6.39635802971029873211511813850, −5.66964589441790450703822438177, −5.12556570591563775340661062282, −4.28214911807671139388102368647, −3.31555313757214129696839985850, −2.55747737322835112196574478489, −1.75725197869287441793468663826, 0,
1.75725197869287441793468663826, 2.55747737322835112196574478489, 3.31555313757214129696839985850, 4.28214911807671139388102368647, 5.12556570591563775340661062282, 5.66964589441790450703822438177, 6.39635802971029873211511813850, 7.30130387924704141598702750688, 7.80789867841411201324819395898
