| L(s) = 1 | − 3-s + 4·4-s + 5·5-s − 7·7-s − 8·9-s − 13·11-s − 4·12-s + 19·13-s − 5·15-s + 16·16-s − 29·17-s + 20·20-s + 7·21-s + 25·25-s + 17·27-s − 28·28-s + 23·29-s + 13·33-s − 35·35-s − 32·36-s − 19·39-s − 52·44-s − 40·45-s + 31·47-s − 16·48-s + 49·49-s + 29·51-s + ⋯ |
| L(s) = 1 | − 1/3·3-s + 4-s + 5-s − 7-s − 8/9·9-s − 1.18·11-s − 1/3·12-s + 1.46·13-s − 1/3·15-s + 16-s − 1.70·17-s + 20-s + 1/3·21-s + 25-s + 0.629·27-s − 28-s + 0.793·29-s + 0.393·33-s − 35-s − 8/9·36-s − 0.487·39-s − 1.18·44-s − 8/9·45-s + 0.659·47-s − 1/3·48-s + 49-s + 0.568·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.105247363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.105247363\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( 1 + T + p^{2} T^{2} \) |
| 11 | \( 1 + 13 T + p^{2} T^{2} \) |
| 13 | \( 1 - 19 T + p^{2} T^{2} \) |
| 17 | \( 1 + 29 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 23 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 - 31 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( 1 - 2 T + p^{2} T^{2} \) |
| 73 | \( 1 - 34 T + p^{2} T^{2} \) |
| 79 | \( 1 + 157 T + p^{2} T^{2} \) |
| 83 | \( 1 + 86 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 149 T + p^{2} T^{2} \) |
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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
−16.22571367389526088572371160974, −15.50734531068932030982196906775, −13.74829341330712259069062381893, −12.82090035080032729313700393887, −11.19546439667118711366527356673, −10.36380462801179566015098963659, −8.696522785147361742116926009064, −6.62398301901393176790011530776, −5.77666833124057227570680820871, −2.70325863282976432569583490987,
2.70325863282976432569583490987, 5.77666833124057227570680820871, 6.62398301901393176790011530776, 8.696522785147361742116926009064, 10.36380462801179566015098963659, 11.19546439667118711366527356673, 12.82090035080032729313700393887, 13.74829341330712259069062381893, 15.50734531068932030982196906775, 16.22571367389526088572371160974
