| L(s) = 1 | − 3-s + 4-s − 12-s − 13-s + 16-s + 2·19-s + 25-s + 27-s + 39-s − 47-s − 48-s + 49-s − 52-s + 2·53-s − 2·57-s − 59-s + 64-s − 73-s − 75-s + 2·76-s − 79-s − 81-s + 2·89-s + 100-s − 101-s − 107-s + 108-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 12-s − 13-s + 16-s + 2·19-s + 25-s + 27-s + 39-s − 47-s − 48-s + 49-s − 52-s + 2·53-s − 2·57-s − 59-s + 64-s − 73-s − 75-s + 2·76-s − 79-s − 81-s + 2·89-s + 100-s − 101-s − 107-s + 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.039848040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039848040\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2003 | \( 1+O(T) \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.547203856879411352392428338647, −8.503257397060096186984408733732, −7.39700807366762124737223152501, −7.09406759402546510588313566854, −6.10866278119992713978791412268, −5.44853423668388818127358941541, −4.79211807291355011649251300726, −3.32768092495314160184024591005, −2.52248101478274207744844407007, −1.10330655657876388817377112587,
1.10330655657876388817377112587, 2.52248101478274207744844407007, 3.32768092495314160184024591005, 4.79211807291355011649251300726, 5.44853423668388818127358941541, 6.10866278119992713978791412268, 7.09406759402546510588313566854, 7.39700807366762124737223152501, 8.503257397060096186984408733732, 9.547203856879411352392428338647
