| L(s) = 1 | − 2-s − 7-s + 8-s + 9-s − 2·11-s + 14-s − 16-s − 18-s + 2·22-s + 23-s + 25-s + 29-s + 37-s − 43-s − 46-s + 49-s − 50-s − 53-s − 56-s − 58-s − 63-s + 64-s + 2·67-s + 71-s + 72-s − 74-s + 2·77-s + ⋯ |
| L(s) = 1 | − 2-s − 7-s + 8-s + 9-s − 2·11-s + 14-s − 16-s − 18-s + 2·22-s + 23-s + 25-s + 29-s + 37-s − 43-s − 46-s + 49-s − 50-s − 53-s − 56-s − 58-s − 63-s + 64-s + 2·67-s + 71-s + 72-s − 74-s + 2·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5208637921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5208637921\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 + T )^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.482721435348213629570169264334, −8.554513802484796327171731539630, −7.897703535298147186120614035021, −7.18039252864529602902467907787, −6.49762536868650824522233983668, −5.17710524920520293304231624057, −4.63040178605986522361461475511, −3.33267995420614926597008141726, −2.34929078094694793167158853150, −0.823817223641608017608410817929,
0.823817223641608017608410817929, 2.34929078094694793167158853150, 3.33267995420614926597008141726, 4.63040178605986522361461475511, 5.17710524920520293304231624057, 6.49762536868650824522233983668, 7.18039252864529602902467907787, 7.897703535298147186120614035021, 8.554513802484796327171731539630, 9.482721435348213629570169264334
