| L(s) = 1 | + 3·2-s + 5·4-s + 7·7-s + 3·8-s + 9·9-s − 6·11-s + 21·14-s − 11·16-s + 27·18-s − 18·22-s − 18·23-s + 35·28-s − 54·29-s − 45·32-s + 45·36-s + 38·37-s − 58·43-s − 30·44-s − 54·46-s + 49·49-s + 6·53-s + 21·56-s − 162·58-s + 63·63-s − 91·64-s + 118·67-s + 114·71-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 5/4·4-s + 7-s + 3/8·8-s + 9-s − 0.545·11-s + 3/2·14-s − 0.687·16-s + 3/2·18-s − 0.818·22-s − 0.782·23-s + 5/4·28-s − 1.86·29-s − 1.40·32-s + 5/4·36-s + 1.02·37-s − 1.34·43-s − 0.681·44-s − 1.17·46-s + 49-s + 6/53·53-s + 3/8·56-s − 2.79·58-s + 63-s − 1.42·64-s + 1.76·67-s + 1.60·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.340644829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.340644829\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{2} T^{2} \) |
| 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 + 6 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 18 T + p^{2} T^{2} \) |
| 29 | \( 1 + 54 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 38 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 58 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 6 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 118 T + p^{2} T^{2} \) |
| 71 | \( 1 - 114 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 94 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−12.73499309097954401241261364186, −11.72114946429536415960283641224, −10.90396618488062604506571068582, −9.634744201767806553925033946699, −8.094351705474056676734448544284, −7.03718216062352377145727309085, −5.70021710145192853096878990643, −4.75622814773392679209724927762, −3.78207417059432049215665954029, −2.05341769902145376345960214580,
2.05341769902145376345960214580, 3.78207417059432049215665954029, 4.75622814773392679209724927762, 5.70021710145192853096878990643, 7.03718216062352377145727309085, 8.094351705474056676734448544284, 9.634744201767806553925033946699, 10.90396618488062604506571068582, 11.72114946429536415960283641224, 12.73499309097954401241261364186
