L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.30 − 0.541i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (1.30 + 0.541i)10-s − 1.00·16-s − 1.00·18-s + (0.541 + 1.30i)20-s + (0.707 − 0.707i)25-s + (1.30 − 0.541i)29-s + (−0.707 − 0.707i)32-s + (−0.707 − 0.707i)36-s + (−0.541 − 1.30i)37-s + (−0.541 + 1.30i)40-s + (−1.30 − 0.541i)41-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.30 − 0.541i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (1.30 + 0.541i)10-s − 1.00·16-s − 1.00·18-s + (0.541 + 1.30i)20-s + (0.707 − 0.707i)25-s + (1.30 − 0.541i)29-s + (−0.707 − 0.707i)32-s + (−0.707 − 0.707i)36-s + (−0.541 − 1.30i)37-s + (−0.541 + 1.30i)40-s + (−1.30 − 0.541i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.707194583\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.707194583\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \) |
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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
−10.08591467227689503087551101827, −9.067655234085041760830647378905, −8.531666519193108901597068288427, −7.61806593987518484356592469388, −6.58640448322124262982557829053, −5.76981125303707214467429050962, −5.26224424715730027274936228948, −4.37161779233012424439630715427, −2.97284965199853163944970200054, −2.00114764386466823104203924190,
1.49818981190896659507726285148, 2.68459072940010737903354980328, 3.33991680556616519864203997718, 4.70566323157587349048432944680, 5.59975823392301793001808203808, 6.30774194116715655176213671600, 6.87950369622594814077753712615, 8.518216317290326086649521261079, 9.257180468459332813226177020899, 10.13481305311738575195396690354