L(s) = 1 | + 3-s + 2.79·5-s + 4.56·7-s + 9-s + 2.91·11-s − 0.850·13-s + 2.79·15-s − 7.79·17-s + 2.29·19-s + 4.56·21-s + 2.74·23-s + 2.82·25-s + 27-s − 6.57·29-s − 7.06·31-s + 2.91·33-s + 12.7·35-s + 8.35·37-s − 0.850·39-s + 4.96·41-s + 11.4·43-s + 2.79·45-s + 12.1·47-s + 13.8·49-s − 7.79·51-s − 6.03·53-s + 8.14·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.25·5-s + 1.72·7-s + 0.333·9-s + 0.877·11-s − 0.235·13-s + 0.722·15-s − 1.89·17-s + 0.527·19-s + 0.996·21-s + 0.572·23-s + 0.564·25-s + 0.192·27-s − 1.22·29-s − 1.26·31-s + 0.506·33-s + 2.15·35-s + 1.37·37-s − 0.136·39-s + 0.776·41-s + 1.74·43-s + 0.416·45-s + 1.77·47-s + 1.97·49-s − 1.09·51-s − 0.829·53-s + 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.937990482\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.937990482\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 + 0.850T + 13T^{2} \) |
| 17 | \( 1 + 7.79T + 17T^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 + 7.06T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 - 4.96T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 6.03T + 53T^{2} \) |
| 59 | \( 1 + 3.64T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 0.961T + 71T^{2} \) |
| 73 | \( 1 + 6.57T + 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 + 8.44T + 97T^{2} \) |
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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
−8.699619627819732285598696385970, −7.58117272003058549176881812029, −7.23327553994034796920022271347, −6.10091924071420412898992365950, −5.53276751763958895779978799688, −4.55188176459474311804986363569, −4.08952895469592253572842420893, −2.59369228695316826599127082913, −1.98917423645848530751150407957, −1.26443149533723587801158508568,
1.26443149533723587801158508568, 1.98917423645848530751150407957, 2.59369228695316826599127082913, 4.08952895469592253572842420893, 4.55188176459474311804986363569, 5.53276751763958895779978799688, 6.10091924071420412898992365950, 7.23327553994034796920022271347, 7.58117272003058549176881812029, 8.699619627819732285598696385970