| L(s) = 1 | − 2-s − 1.27·3-s + 4-s + 5-s + 1.27·6-s − 3.95·7-s − 8-s − 1.38·9-s − 10-s + 4.66·11-s − 1.27·12-s + 13-s + 3.95·14-s − 1.27·15-s + 16-s − 1.48·17-s + 1.38·18-s + 3.80·19-s + 20-s + 5.02·21-s − 4.66·22-s − 7.81·23-s + 1.27·24-s + 25-s − 26-s + 5.57·27-s − 3.95·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.733·3-s + 0.5·4-s + 0.447·5-s + 0.518·6-s − 1.49·7-s − 0.353·8-s − 0.462·9-s − 0.316·10-s + 1.40·11-s − 0.366·12-s + 0.277·13-s + 1.05·14-s − 0.327·15-s + 0.250·16-s − 0.360·17-s + 0.326·18-s + 0.872·19-s + 0.223·20-s + 1.09·21-s − 0.994·22-s − 1.62·23-s + 0.259·24-s + 0.200·25-s − 0.196·26-s + 1.07·27-s − 0.747·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 1.27T + 3T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 + 7.81T + 23T^{2} \) |
| 29 | \( 1 - 0.373T + 29T^{2} \) |
| 37 | \( 1 + 3.50T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 3.20T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 - 6.57T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 - 9.44T + 71T^{2} \) |
| 73 | \( 1 - 3.09T + 73T^{2} \) |
| 79 | \( 1 - 0.814T + 79T^{2} \) |
| 83 | \( 1 + 7.28T + 83T^{2} \) |
| 89 | \( 1 + 6.30T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.280989349555940944451010693158, −6.99371918317170394815926957831, −6.70754238731445712385645531489, −5.92288943255439808004382840459, −5.55211834974132933357500369066, −4.09976084253446387767770664071, −3.36561641229147613151586329518, −2.35142211570029810164769277667, −1.12174068118413922219179234815, 0,
1.12174068118413922219179234815, 2.35142211570029810164769277667, 3.36561641229147613151586329518, 4.09976084253446387767770664071, 5.55211834974132933357500369066, 5.92288943255439808004382840459, 6.70754238731445712385645531489, 6.99371918317170394815926957831, 8.280989349555940944451010693158
