| L(s) = 1 | + 5·3-s + 4·4-s + 16·9-s − 11·11-s + 20·12-s + 16·16-s − 35·23-s + 35·27-s − 37·31-s − 55·33-s + 64·36-s + 25·37-s − 44·44-s − 50·47-s + 80·48-s + 49·49-s + 70·53-s + 107·59-s + 64·64-s − 35·67-s − 175·69-s − 133·71-s + 31·81-s − 97·89-s − 140·92-s − 185·93-s − 95·97-s + ⋯ |
| L(s) = 1 | + 5/3·3-s + 4-s + 16/9·9-s − 11-s + 5/3·12-s + 16-s − 1.52·23-s + 1.29·27-s − 1.19·31-s − 5/3·33-s + 16/9·36-s + 0.675·37-s − 44-s − 1.06·47-s + 5/3·48-s + 49-s + 1.32·53-s + 1.81·59-s + 64-s − 0.522·67-s − 2.53·69-s − 1.87·71-s + 0.382·81-s − 1.08·89-s − 1.52·92-s − 1.98·93-s − 0.979·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.200741775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.200741775\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( 1 - 5 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 35 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 37 T + p^{2} T^{2} \) |
| 37 | \( 1 - 25 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 50 T + p^{2} T^{2} \) |
| 53 | \( 1 - 70 T + p^{2} T^{2} \) |
| 59 | \( 1 - 107 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 + 35 T + p^{2} T^{2} \) |
| 71 | \( 1 + 133 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 97 T + p^{2} T^{2} \) |
| 97 | \( 1 + 95 T + p^{2} T^{2} \) |
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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
−11.70252244653229272988728789552, −10.49713713700157037082299522380, −9.799179467378117947108662290380, −8.589690281052869912569493096174, −7.82732655977153555089795883751, −7.11980403834502415525552504414, −5.70224133999946930035905387177, −3.94841571058193275478169853868, −2.81542248842016306446563241604, −1.94808752404925098621157871892,
1.94808752404925098621157871892, 2.81542248842016306446563241604, 3.94841571058193275478169853868, 5.70224133999946930035905387177, 7.11980403834502415525552504414, 7.82732655977153555089795883751, 8.589690281052869912569493096174, 9.799179467378117947108662290380, 10.49713713700157037082299522380, 11.70252244653229272988728789552
