| L(s) = 1 | + 2-s − 7-s − 8-s + 9-s − 11-s − 14-s − 16-s + 18-s − 22-s + 23-s − 29-s + 37-s + 43-s + 46-s + 49-s − 2·53-s + 56-s − 58-s − 63-s + 64-s + 67-s − 71-s − 72-s + 74-s + 77-s − 79-s + 81-s + ⋯ |
| L(s) = 1 | + 2-s − 7-s − 8-s + 9-s − 11-s − 14-s − 16-s + 18-s − 22-s + 23-s − 29-s + 37-s + 43-s + 46-s + 49-s − 2·53-s + 56-s − 58-s − 63-s + 64-s + 67-s − 71-s − 72-s + 74-s + 77-s − 79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8869601442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8869601442\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 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 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 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
−12.90962454210898858484624780878, −12.55882728637175638347345030269, −11.11325725636170649079381317752, −9.931651058410106395609982969638, −9.120960691540419640056208813700, −7.57820558602448297607061133246, −6.41858029876641888947121384092, −5.28850110961092873256574941658, −4.13099145489053729897691030505, −2.89897822964866581003696244265,
2.89897822964866581003696244265, 4.13099145489053729897691030505, 5.28850110961092873256574941658, 6.41858029876641888947121384092, 7.57820558602448297607061133246, 9.120960691540419640056208813700, 9.931651058410106395609982969638, 11.11325725636170649079381317752, 12.55882728637175638347345030269, 12.90962454210898858484624780878
