| L(s) = 1 | + 2-s + 1.95·3-s + 4-s − 5-s + 1.95·6-s + 8-s + 0.825·9-s − 10-s − 1.60·11-s + 1.95·12-s − 3.29·13-s − 1.95·15-s + 16-s + 17-s + 0.825·18-s + 3.56·19-s − 20-s − 1.60·22-s − 3.26·23-s + 1.95·24-s + 25-s − 3.29·26-s − 4.25·27-s − 5.08·29-s − 1.95·30-s − 9.61·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.12·3-s + 0.5·4-s − 0.447·5-s + 0.798·6-s + 0.353·8-s + 0.275·9-s − 0.316·10-s − 0.484·11-s + 0.564·12-s − 0.914·13-s − 0.505·15-s + 0.250·16-s + 0.242·17-s + 0.194·18-s + 0.817·19-s − 0.223·20-s − 0.342·22-s − 0.681·23-s + 0.399·24-s + 0.200·25-s − 0.646·26-s − 0.818·27-s − 0.944·29-s − 0.357·30-s − 1.72·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.95T + 3T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 + 9.61T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 + 2.59T + 41T^{2} \) |
| 43 | \( 1 - 3.35T + 43T^{2} \) |
| 47 | \( 1 - 2.15T + 47T^{2} \) |
| 53 | \( 1 + 0.907T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6.15T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 5.80T + 71T^{2} \) |
| 73 | \( 1 + 6.30T + 73T^{2} \) |
| 79 | \( 1 - 1.59T + 79T^{2} \) |
| 83 | \( 1 + 2.40T + 83T^{2} \) |
| 89 | \( 1 - 2.55T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.50605750506920593111512862880, −7.05532361386309078035359876349, −5.84798806118344963019187864542, −5.36937952080170245021722344637, −4.51440969486547773558241349754, −3.69992011804601892403314130113, −3.20399342178242729481708953837, −2.43692705342144904123460976246, −1.68969413653695537881048461222, 0,
1.68969413653695537881048461222, 2.43692705342144904123460976246, 3.20399342178242729481708953837, 3.69992011804601892403314130113, 4.51440969486547773558241349754, 5.36937952080170245021722344637, 5.84798806118344963019187864542, 7.05532361386309078035359876349, 7.50605750506920593111512862880
