| L(s) = 1 | − 1.23·3-s + 0.236·7-s − 1.47·9-s + 11-s + 2.23·13-s − 2.47·17-s + 19-s − 0.291·21-s − 23-s + 5.52·27-s − 6.23·29-s − 8.47·31-s − 1.23·33-s + 6.76·37-s − 2.76·39-s + 11.9·41-s + 11.4·43-s − 1.70·47-s − 6.94·49-s + 3.05·51-s − 3.23·53-s − 1.23·57-s − 1.23·59-s − 2.76·61-s − 0.347·63-s − 4.94·67-s + 1.23·69-s + ⋯ |
| L(s) = 1 | − 0.713·3-s + 0.0892·7-s − 0.490·9-s + 0.301·11-s + 0.620·13-s − 0.599·17-s + 0.229·19-s − 0.0636·21-s − 0.208·23-s + 1.06·27-s − 1.15·29-s − 1.52·31-s − 0.215·33-s + 1.11·37-s − 0.442·39-s + 1.86·41-s + 1.74·43-s − 0.249·47-s − 0.992·49-s + 0.427·51-s − 0.444·53-s − 0.163·57-s − 0.160·59-s − 0.353·61-s − 0.0437·63-s − 0.604·67-s + 0.148·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.200941399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.200941399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + 3.23T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 0.527T + 73T^{2} \) |
| 79 | \( 1 + 7.18T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.259327125608239487620625767939, −7.58497734860849402406955210640, −6.77848240889571913938302852737, −5.89743284607374795219123428212, −5.68465014782805229123921118187, −4.59796293739742178973279023073, −3.89908739598071179764982796108, −2.90358027288282905760693584927, −1.84145089025482935113419135909, −0.62910679321861749837331227631,
0.62910679321861749837331227631, 1.84145089025482935113419135909, 2.90358027288282905760693584927, 3.89908739598071179764982796108, 4.59796293739742178973279023073, 5.68465014782805229123921118187, 5.89743284607374795219123428212, 6.77848240889571913938302852737, 7.58497734860849402406955210640, 8.259327125608239487620625767939
