| L(s) = 1 | + 3-s − 7-s − 2·9-s − 11-s − 13-s − 3·17-s − 4·19-s − 21-s + 2·23-s − 5·27-s − 29-s − 6·31-s − 33-s + 2·37-s − 39-s − 10·41-s + 9·47-s + 49-s − 3·51-s − 14·53-s − 4·57-s + 6·59-s − 4·61-s + 2·63-s + 10·67-s + 2·69-s − 16·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 0.727·17-s − 0.917·19-s − 0.218·21-s + 0.417·23-s − 0.962·27-s − 0.185·29-s − 1.07·31-s − 0.174·33-s + 0.328·37-s − 0.160·39-s − 1.56·41-s + 1.31·47-s + 1/7·49-s − 0.420·51-s − 1.92·53-s − 0.529·57-s + 0.781·59-s − 0.512·61-s + 0.251·63-s + 1.22·67-s + 0.240·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−9.027303773761236255503955761867, −8.508598615719724580818612147870, −7.62019651371471199947336815910, −6.75268223668308908312188932664, −5.88222710772062282467953861310, −4.92848758497648143896463525043, −3.84301196332857562899953965920, −2.90638102415243745513766468416, −2.00113560337988224809643036481, 0,
2.00113560337988224809643036481, 2.90638102415243745513766468416, 3.84301196332857562899953965920, 4.92848758497648143896463525043, 5.88222710772062282467953861310, 6.75268223668308908312188932664, 7.62019651371471199947336815910, 8.508598615719724580818612147870, 9.027303773761236255503955761867
