| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s − 2·11-s + 2·13-s − 15-s + 4·17-s − 4·19-s + 2·21-s − 4·23-s + 25-s − 27-s + 2·29-s + 4·31-s + 2·33-s − 2·35-s − 2·37-s − 2·39-s − 6·41-s + 4·43-s + 45-s + 8·47-s − 3·49-s − 4·51-s − 10·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s + 0.970·17-s − 0.917·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.348·33-s − 0.338·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.560·51-s − 1.37·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.131072938444719093648071025865, −7.31276043143988012921852589591, −6.39981137929568018254821286117, −6.02068627898151325117075778591, −5.25126839273566821455291492827, −4.34561655643917364661199646559, −3.42583354357075403934039312397, −2.50530685019232098024851380185, −1.33842132332385794180824115855, 0,
1.33842132332385794180824115855, 2.50530685019232098024851380185, 3.42583354357075403934039312397, 4.34561655643917364661199646559, 5.25126839273566821455291492827, 6.02068627898151325117075778591, 6.39981137929568018254821286117, 7.31276043143988012921852589591, 8.131072938444719093648071025865
