L(s) = 1 | − 1.40·3-s + 5-s − 2.17·7-s − 1.01·9-s − 5.73·11-s − 1.40·15-s − 5.24·17-s − 7.00·19-s + 3.05·21-s − 8.19·23-s + 25-s + 5.65·27-s + 9.54·29-s − 2.41·31-s + 8.07·33-s − 2.17·35-s + 7.16·37-s + 7.16·41-s + 7.76·43-s − 1.01·45-s − 5.41·47-s − 2.28·49-s + 7.38·51-s + 3.94·53-s − 5.73·55-s + 9.87·57-s − 0.993·59-s + ⋯ |
L(s) = 1 | − 0.813·3-s + 0.447·5-s − 0.820·7-s − 0.338·9-s − 1.72·11-s − 0.363·15-s − 1.27·17-s − 1.60·19-s + 0.667·21-s − 1.70·23-s + 0.200·25-s + 1.08·27-s + 1.77·29-s − 0.434·31-s + 1.40·33-s − 0.366·35-s + 1.17·37-s + 1.11·41-s + 1.18·43-s − 0.151·45-s − 0.790·47-s − 0.326·49-s + 1.03·51-s + 0.541·53-s − 0.773·55-s + 1.30·57-s − 0.129·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4727069839\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4727069839\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 1.40T + 3T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 - 9.54T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 - 7.16T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 - 3.94T + 53T^{2} \) |
| 59 | \( 1 + 0.993T + 59T^{2} \) |
| 61 | \( 1 + 4.28T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 + 3.06T + 73T^{2} \) |
| 79 | \( 1 - 7.33T + 79T^{2} \) |
| 83 | \( 1 - 2.86T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 7.35T + 97T^{2} \) |
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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
−8.459748789269108988459180007840, −8.038828941503175546397056878197, −6.83978540725370831230461817197, −6.20871363875079237521290160436, −5.81643194647875426476296821287, −4.85111170569980478952357256394, −4.15431275607598688754660123102, −2.72141173595838055066811673341, −2.29057776353970211763642506790, −0.39141242794799086753314643804,
0.39141242794799086753314643804, 2.29057776353970211763642506790, 2.72141173595838055066811673341, 4.15431275607598688754660123102, 4.85111170569980478952357256394, 5.81643194647875426476296821287, 6.20871363875079237521290160436, 6.83978540725370831230461817197, 8.038828941503175546397056878197, 8.459748789269108988459180007840