| L(s) = 1 | + 3.31·3-s + 4·4-s + 2·9-s + 11·11-s + 13.2·12-s + 16·16-s − 29.8·23-s − 23.2·27-s + 37·31-s + 36.4·33-s + 8·36-s − 69.6·37-s + 44·44-s + 79.5·47-s + 53.0·48-s + 49·49-s − 79.5·53-s − 107·59-s + 64·64-s + 129.·67-s − 99·69-s − 133·71-s − 94.9·81-s − 97·89-s − 119.·92-s + 122.·93-s − 169.·97-s + ⋯ |
| L(s) = 1 | + 1.10·3-s + 4-s + 0.222·9-s + 11-s + 1.10·12-s + 16-s − 1.29·23-s − 0.859·27-s + 1.19·31-s + 1.10·33-s + 0.222·36-s − 1.88·37-s + 44-s + 1.69·47-s + 1.10·48-s + 0.999·49-s − 1.50·53-s − 1.81·59-s + 64-s + 1.93·67-s − 1.43·69-s − 1.87·71-s − 1.17·81-s − 1.08·89-s − 1.29·92-s + 1.31·93-s − 1.74·97-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.823193311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.823193311\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 4T^{2} \) |
| 3 | \( 1 - 3.31T + 9T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 29.8T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 37T + 961T^{2} \) |
| 37 | \( 1 + 69.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 79.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 79.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 107T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 129.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 133T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 97T + 7.92e3T^{2} \) |
| 97 | \( 1 + 169.T + 9.40e3T^{2} \) |
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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
−11.79177693603448769702976386782, −10.69648386059242525717558876116, −9.698076997253849789726788020157, −8.698221035463996934627816353112, −7.85013072071112084194087207326, −6.84550245497527454405280617778, −5.83218345298344293033893227129, −4.00877688340984973565850450219, −2.92009732547528960577531915043, −1.74066615048293645852928406887,
1.74066615048293645852928406887, 2.92009732547528960577531915043, 4.00877688340984973565850450219, 5.83218345298344293033893227129, 6.84550245497527454405280617778, 7.85013072071112084194087207326, 8.698221035463996934627816353112, 9.698076997253849789726788020157, 10.69648386059242525717558876116, 11.79177693603448769702976386782
