L(s) = 1 | − 1.39·3-s − 3.68·5-s + 0.935·7-s − 1.05·9-s − 11-s − 0.935·13-s + 5.13·15-s + 5.08·17-s − 19-s − 1.30·21-s − 0.699·23-s + 8.60·25-s + 5.65·27-s + 10.3·29-s − 1.63·31-s + 1.39·33-s − 3.45·35-s + 3.78·37-s + 1.30·39-s + 3.35·41-s − 5.45·43-s + 3.90·45-s + 0.658·47-s − 6.12·49-s − 7.08·51-s − 3.54·53-s + 3.68·55-s + ⋯ |
L(s) = 1 | − 0.804·3-s − 1.64·5-s + 0.353·7-s − 0.352·9-s − 0.301·11-s − 0.259·13-s + 1.32·15-s + 1.23·17-s − 0.229·19-s − 0.284·21-s − 0.145·23-s + 1.72·25-s + 1.08·27-s + 1.92·29-s − 0.293·31-s + 0.242·33-s − 0.583·35-s + 0.621·37-s + 0.208·39-s + 0.524·41-s − 0.831·43-s + 0.581·45-s + 0.0961·47-s − 0.874·49-s − 0.991·51-s − 0.486·53-s + 0.497·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 7 | \( 1 - 0.935T + 7T^{2} \) |
| 13 | \( 1 + 0.935T + 13T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 23 | \( 1 + 0.699T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 3.78T + 37T^{2} \) |
| 41 | \( 1 - 3.35T + 41T^{2} \) |
| 43 | \( 1 + 5.45T + 43T^{2} \) |
| 47 | \( 1 - 0.658T + 47T^{2} \) |
| 53 | \( 1 + 3.54T + 53T^{2} \) |
| 59 | \( 1 + 8.23T + 59T^{2} \) |
| 61 | \( 1 - 2.14T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 6.67T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 + 7.41T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 2.22T + 97T^{2} \) |
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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
−8.053466214792162748751247827279, −7.70416616273240383518077122937, −6.76323490227082620506804662079, −5.99884070084374497574240546242, −5.01172175451105415746864038396, −4.57235371442327165379009443948, −3.53371235132391945990980449805, −2.77582724676706233568486880840, −1.06075971026539627873777655255, 0,
1.06075971026539627873777655255, 2.77582724676706233568486880840, 3.53371235132391945990980449805, 4.57235371442327165379009443948, 5.01172175451105415746864038396, 5.99884070084374497574240546242, 6.76323490227082620506804662079, 7.70416616273240383518077122937, 8.053466214792162748751247827279