L(s) = 1 | + 3-s + 2·7-s − 2·9-s + 2·11-s − 2·13-s − 5·17-s + 4·19-s + 2·21-s + 6·23-s − 5·25-s − 5·27-s − 3·31-s + 2·33-s + 2·37-s − 2·39-s − 4·41-s + 6·43-s + 9·47-s − 3·49-s − 5·51-s − 53-s + 4·57-s + 6·59-s + 13·61-s − 4·63-s + 15·67-s + 6·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.603·11-s − 0.554·13-s − 1.21·17-s + 0.917·19-s + 0.436·21-s + 1.25·23-s − 25-s − 0.962·27-s − 0.538·31-s + 0.348·33-s + 0.328·37-s − 0.320·39-s − 0.624·41-s + 0.914·43-s + 1.31·47-s − 3/7·49-s − 0.700·51-s − 0.137·53-s + 0.529·57-s + 0.781·59-s + 1.66·61-s − 0.503·63-s + 1.83·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.556921677\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.556921677\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−7.60021948092764537632079419322, −7.21899013653519796570120960038, −6.36071595459821884962031025332, −5.52281653013098876677624717854, −4.96377522191538833077622427597, −4.12507235348094948962973805824, −3.42411867458366080974259746990, −2.50599320209037879567578672909, −1.92447861180603306823598813116, −0.73169470468344138877721610044,
0.73169470468344138877721610044, 1.92447861180603306823598813116, 2.50599320209037879567578672909, 3.42411867458366080974259746990, 4.12507235348094948962973805824, 4.96377522191538833077622427597, 5.52281653013098876677624717854, 6.36071595459821884962031025332, 7.21899013653519796570120960038, 7.60021948092764537632079419322