L(s) = 1 | + 0.423·2-s − 1.82·4-s + 4.20·5-s + 2.99·7-s − 1.61·8-s + 1.77·10-s + 2.55·11-s + 2.04·13-s + 1.26·14-s + 2.95·16-s − 6.75·17-s + 1.19·19-s − 7.65·20-s + 1.08·22-s + 1.18·23-s + 12.6·25-s + 0.864·26-s − 5.45·28-s + 7.94·29-s − 5.64·31-s + 4.48·32-s − 2.85·34-s + 12.5·35-s − 8.25·37-s + 0.506·38-s − 6.79·40-s − 3.91·41-s + ⋯ |
L(s) = 1 | + 0.299·2-s − 0.910·4-s + 1.88·5-s + 1.13·7-s − 0.571·8-s + 0.562·10-s + 0.769·11-s + 0.566·13-s + 0.338·14-s + 0.739·16-s − 1.63·17-s + 0.274·19-s − 1.71·20-s + 0.230·22-s + 0.246·23-s + 2.53·25-s + 0.169·26-s − 1.03·28-s + 1.47·29-s − 1.01·31-s + 0.792·32-s − 0.490·34-s + 2.12·35-s − 1.35·37-s + 0.0821·38-s − 1.07·40-s − 0.610·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.558625787\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.558625787\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 2 | \( 1 - 0.423T + 2T^{2} \) |
| 5 | \( 1 - 4.20T + 5T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 19 | \( 1 - 1.19T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 + 5.64T + 31T^{2} \) |
| 37 | \( 1 + 8.25T + 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 5.08T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 4.66T + 67T^{2} \) |
| 71 | \( 1 - 8.01T + 71T^{2} \) |
| 73 | \( 1 + 8.75T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 6.19T + 83T^{2} \) |
| 89 | \( 1 - 8.51T + 89T^{2} \) |
| 97 | \( 1 + 5.18T + 97T^{2} \) |
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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
−9.547425027342224580170977968709, −8.718398125407190863421427256618, −8.512457963796442058587555799523, −6.83361225385016482774613046415, −6.24894640527778319736360840406, −5.16006268335232417359936423055, −4.87260149318089619587155488882, −3.62894487964324394709631683888, −2.19882339911021409894127224612, −1.30153744496042672568171782968,
1.30153744496042672568171782968, 2.19882339911021409894127224612, 3.62894487964324394709631683888, 4.87260149318089619587155488882, 5.16006268335232417359936423055, 6.24894640527778319736360840406, 6.83361225385016482774613046415, 8.512457963796442058587555799523, 8.718398125407190863421427256618, 9.547425027342224580170977968709