| L(s) = 1 | + 2.18·2-s + 2.15·3-s + 2.77·4-s − 3.43·5-s + 4.71·6-s − 3.93·7-s + 1.69·8-s + 1.65·9-s − 7.49·10-s + 5.98·12-s − 3.31·13-s − 8.60·14-s − 7.40·15-s − 1.84·16-s − 2.80·17-s + 3.61·18-s + 19-s − 9.52·20-s − 8.50·21-s + 6.88·23-s + 3.65·24-s + 6.77·25-s − 7.23·26-s − 2.90·27-s − 10.9·28-s − 5.67·29-s − 16.1·30-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.24·3-s + 1.38·4-s − 1.53·5-s + 1.92·6-s − 1.48·7-s + 0.598·8-s + 0.551·9-s − 2.37·10-s + 1.72·12-s − 0.918·13-s − 2.30·14-s − 1.91·15-s − 0.462·16-s − 0.680·17-s + 0.852·18-s + 0.229·19-s − 2.12·20-s − 1.85·21-s + 1.43·23-s + 0.746·24-s + 1.35·25-s − 1.41·26-s − 0.558·27-s − 2.06·28-s − 1.05·29-s − 2.95·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 + 6.39T + 37T^{2} \) |
| 41 | \( 1 + 0.560T + 41T^{2} \) |
| 43 | \( 1 - 9.40T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 - 4.35T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 + 9.95T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 8.95T + 73T^{2} \) |
| 79 | \( 1 + 8.49T + 79T^{2} \) |
| 83 | \( 1 - 5.21T + 83T^{2} \) |
| 89 | \( 1 + 7.28T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.708252831078768243067101776499, −7.47983343273724032187546471655, −7.17879592643209849155964299813, −6.30972510714767163581102130826, −5.16090888062739913036023286989, −4.30411285662515979332953155986, −3.55714650059919658548916379749, −3.15382659376182640238487585044, −2.42747140967625438797621191451, 0,
2.42747140967625438797621191451, 3.15382659376182640238487585044, 3.55714650059919658548916379749, 4.30411285662515979332953155986, 5.16090888062739913036023286989, 6.30972510714767163581102130826, 7.17879592643209849155964299813, 7.47983343273724032187546471655, 8.708252831078768243067101776499
