L(s) = 1 | + 3-s + 3.42·7-s + 9-s + 2.71·11-s + 13-s − 1.08·17-s − 0.623·19-s + 3.42·21-s + 3.08·23-s + 27-s − 3.69·29-s + 6.65·31-s + 2.71·33-s + 5.89·37-s + 39-s + 7.00·41-s − 11.3·43-s − 2.05·47-s + 4.70·49-s − 1.08·51-s + 13.2·53-s − 0.623·57-s − 2.23·59-s − 7.14·61-s + 3.42·63-s − 9.01·67-s + 3.08·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.29·7-s + 0.333·9-s + 0.819·11-s + 0.277·13-s − 0.262·17-s − 0.143·19-s + 0.746·21-s + 0.642·23-s + 0.192·27-s − 0.685·29-s + 1.19·31-s + 0.473·33-s + 0.969·37-s + 0.160·39-s + 1.09·41-s − 1.73·43-s − 0.299·47-s + 0.672·49-s − 0.151·51-s + 1.82·53-s − 0.0826·57-s − 0.291·59-s − 0.915·61-s + 0.431·63-s − 1.10·67-s + 0.371·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.543613042\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.543613042\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 + 0.623T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 - 6.65T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 2.05T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 7.14T + 61T^{2} \) |
| 67 | \( 1 + 9.01T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 7.50T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 + 6.73T + 83T^{2} \) |
| 89 | \( 1 - 5.78T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−7.981433772597370853936949045480, −7.25944836187008485745902530731, −6.56618579410049687538560514944, −5.77660304573176916246933357989, −4.85152521492368230279362429966, −4.35034452117160499760157381955, −3.57378449460581450386639525510, −2.60990456274094732348046896348, −1.75934012373145459167738188186, −0.988649707765456700666418385598,
0.988649707765456700666418385598, 1.75934012373145459167738188186, 2.60990456274094732348046896348, 3.57378449460581450386639525510, 4.35034452117160499760157381955, 4.85152521492368230279362429966, 5.77660304573176916246933357989, 6.56618579410049687538560514944, 7.25944836187008485745902530731, 7.981433772597370853936949045480