L(s) = 1 | + 1.53·3-s + 4-s + 1.34·9-s + 1.53·12-s − 1.87·13-s + 16-s − 19-s + 25-s + 0.532·27-s + 1.34·36-s − 2.87·39-s − 1.87·47-s + 1.53·48-s + 49-s − 1.87·52-s − 53-s − 1.53·57-s + 0.347·59-s + 64-s + 0.347·73-s + 1.53·75-s − 76-s + 0.347·79-s − 0.532·81-s − 89-s + 100-s − 1.87·101-s + ⋯ |
L(s) = 1 | + 1.53·3-s + 4-s + 1.34·9-s + 1.53·12-s − 1.87·13-s + 16-s − 19-s + 25-s + 0.532·27-s + 1.34·36-s − 2.87·39-s − 1.87·47-s + 1.53·48-s + 49-s − 1.87·52-s − 53-s − 1.53·57-s + 0.347·59-s + 64-s + 0.347·73-s + 1.53·75-s − 76-s + 0.347·79-s − 0.532·81-s − 89-s + 100-s − 1.87·101-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.228631311\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.228631311\) |
\(L(1)\) |
|
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 | \( 1 - T^{2} \) |
| 3 | \( 1 - 1.53T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.87T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.87T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - 0.347T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 - 0.347T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.350717474866893376483197923257, −8.428208883916818202770840587223, −7.84498276518176958371490651454, −7.12960199598308512493803321698, −6.54400559867713352674993768852, −5.25384165983110084796513466485, −4.27648195618746220461459441030, −3.14331586431822676243864670141, −2.55635132839001603736068856688, −1.80124379298909506691068598074,
1.80124379298909506691068598074, 2.55635132839001603736068856688, 3.14331586431822676243864670141, 4.27648195618746220461459441030, 5.25384165983110084796513466485, 6.54400559867713352674993768852, 7.12960199598308512493803321698, 7.84498276518176958371490651454, 8.428208883916818202770840587223, 9.350717474866893376483197923257