L(s) = 1 | + 3-s + 1.52·5-s + 3.74·7-s + 9-s + 1.08·11-s + 3.02·13-s + 1.52·15-s − 3.10·17-s + 4.15·19-s + 3.74·21-s − 2.67·25-s + 27-s + 2.66·29-s + 6.80·31-s + 1.08·33-s + 5.71·35-s − 6.84·37-s + 3.02·39-s + 9.95·41-s − 4.00·43-s + 1.52·45-s − 1.56·47-s + 7.04·49-s − 3.10·51-s + 2.86·53-s + 1.66·55-s + 4.15·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.682·5-s + 1.41·7-s + 0.333·9-s + 0.328·11-s + 0.839·13-s + 0.394·15-s − 0.752·17-s + 0.952·19-s + 0.817·21-s − 0.534·25-s + 0.192·27-s + 0.494·29-s + 1.22·31-s + 0.189·33-s + 0.966·35-s − 1.12·37-s + 0.484·39-s + 1.55·41-s − 0.611·43-s + 0.227·45-s − 0.228·47-s + 1.00·49-s − 0.434·51-s + 0.394·53-s + 0.223·55-s + 0.549·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.962831745\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.962831745\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 - 1.08T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 17 | \( 1 + 3.10T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 - 6.80T + 31T^{2} \) |
| 37 | \( 1 + 6.84T + 37T^{2} \) |
| 41 | \( 1 - 9.95T + 41T^{2} \) |
| 43 | \( 1 + 4.00T + 43T^{2} \) |
| 47 | \( 1 + 1.56T + 47T^{2} \) |
| 53 | \( 1 - 2.86T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 3.21T + 73T^{2} \) |
| 79 | \( 1 - 3.09T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 3.83T + 97T^{2} \) |
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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
−8.021970155559343971147410877236, −7.53978362425438653429475936887, −6.57947730510175273929079561314, −5.94875362206347756137195970537, −5.06667844263423562652400731437, −4.47508955057998780861363840157, −3.62594105460020483122440590510, −2.62894505839427235890974994476, −1.78884814285047352506616815109, −1.12944784821482962784186559573,
1.12944784821482962784186559573, 1.78884814285047352506616815109, 2.62894505839427235890974994476, 3.62594105460020483122440590510, 4.47508955057998780861363840157, 5.06667844263423562652400731437, 5.94875362206347756137195970537, 6.57947730510175273929079561314, 7.53978362425438653429475936887, 8.021970155559343971147410877236