| L(s) = 1 | − 2·3-s − 14·7-s − 5·9-s + 50·13-s + 14·19-s + 28·21-s + 28·27-s + 14·31-s − 4·37-s − 100·39-s + 82·43-s + 49·49-s − 28·57-s − 2·61-s + 70·63-s + 34·67-s + 140·73-s + 116·79-s − 11·81-s − 700·91-s − 28·93-s + 98·97-s − 308·103-s − 50·109-s + 8·111-s − 250·117-s + 170·121-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 2·7-s − 5/9·9-s + 3.84·13-s + 0.736·19-s + 4/3·21-s + 1.03·27-s + 0.451·31-s − 0.108·37-s − 2.56·39-s + 1.90·43-s + 49-s − 0.491·57-s − 0.0327·61-s + 10/9·63-s + 0.507·67-s + 1.91·73-s + 1.46·79-s − 0.135·81-s − 7.69·91-s − 0.301·93-s + 1.01·97-s − 2.99·103-s − 0.458·109-s + 0.0720·111-s − 2.13·117-s + 1.40·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.149135297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.149135297\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 170 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 70 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 410 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 118 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 41 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2090 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8282 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2590 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 49 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.563407708653643241170635467357, −9.278182816574455648955918982065, −9.207131210286499654580076862163, −8.402871194311325781588232026964, −8.312442937209102931080426509905, −7.87168571617021974592857055916, −6.99636776066788243783742105982, −6.65988582665227389179627870133, −6.35580454539860151211782630768, −6.09194075330437405713274806655, −5.63317501891314116629611310768, −5.45294969254789258360792834203, −4.51642687810287806640613590115, −3.90038022446454594277232940380, −3.59098456162855855798103301028, −3.19694064246147444481267059699, −2.78290984440048115911877160415, −1.75474013677593469069197409119, −0.864177878335796560575921671162, −0.64794728256991558033707826395,
0.64794728256991558033707826395, 0.864177878335796560575921671162, 1.75474013677593469069197409119, 2.78290984440048115911877160415, 3.19694064246147444481267059699, 3.59098456162855855798103301028, 3.90038022446454594277232940380, 4.51642687810287806640613590115, 5.45294969254789258360792834203, 5.63317501891314116629611310768, 6.09194075330437405713274806655, 6.35580454539860151211782630768, 6.65988582665227389179627870133, 6.99636776066788243783742105982, 7.87168571617021974592857055916, 8.312442937209102931080426509905, 8.402871194311325781588232026964, 9.207131210286499654580076862163, 9.278182816574455648955918982065, 9.563407708653643241170635467357
