| L(s) = 1 | + 4·4-s + 21·11-s − 17·13-s + 16·16-s + 9·17-s − 3·23-s + 25·25-s + 19·31-s − 39·41-s − 43·43-s + 84·44-s + 78·47-s + 49·49-s − 68·52-s − 63·53-s + 54·59-s + 64·64-s + 91·67-s + 36·68-s − 14·79-s − 123·83-s − 12·92-s − 193·97-s + 100·100-s − 159·101-s − 181·103-s − 42·107-s + ⋯ |
| L(s) = 1 | + 4-s + 1.90·11-s − 1.30·13-s + 16-s + 9/17·17-s − 0.130·23-s + 25-s + 0.612·31-s − 0.951·41-s − 43-s + 1.90·44-s + 1.65·47-s + 49-s − 1.30·52-s − 1.18·53-s + 0.915·59-s + 64-s + 1.35·67-s + 9/17·68-s − 0.177·79-s − 1.48·83-s − 0.130·92-s − 1.98·97-s + 100-s − 1.57·101-s − 1.75·103-s − 0.392·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.305088484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.305088484\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 - 21 T + p^{2} T^{2} \) |
| 13 | \( 1 + 17 T + p^{2} T^{2} \) |
| 17 | \( 1 - 9 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 3 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 19 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 + 39 T + p^{2} T^{2} \) |
| 47 | \( 1 - 78 T + p^{2} T^{2} \) |
| 53 | \( 1 + 63 T + p^{2} T^{2} \) |
| 59 | \( 1 - 54 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 91 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 14 T + p^{2} T^{2} \) |
| 83 | \( 1 + 123 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 193 T + p^{2} T^{2} \) |
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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
−11.25044150787781939639381284021, −10.19557427297173655848752306076, −9.412508172947311189847837583100, −8.277140290397387096617856577829, −7.07867103715640404421556001380, −6.61545512344545415078049133458, −5.37609669643299440829738158459, −3.99424623108223917905424003768, −2.71989431861053188899707025193, −1.33151835608797037147591353617,
1.33151835608797037147591353617, 2.71989431861053188899707025193, 3.99424623108223917905424003768, 5.37609669643299440829738158459, 6.61545512344545415078049133458, 7.07867103715640404421556001380, 8.277140290397387096617856577829, 9.412508172947311189847837583100, 10.19557427297173655848752306076, 11.25044150787781939639381284021
