L(s) = 1 | + (−0.382 + 0.923i)2-s + (−1.30 + 1.30i)3-s + (−0.707 − 0.707i)4-s + (−0.707 − 1.70i)6-s + (0.923 − 0.382i)8-s − 2.41i·9-s + 1.41i·11-s + 1.84·12-s + (1.30 + 1.30i)13-s + i·16-s + (2.23 + 0.923i)18-s + 19-s + (−1.30 − 0.541i)22-s + (−0.707 + 1.70i)24-s + (−1.70 + 0.707i)26-s + (1.84 + 1.84i)27-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)2-s + (−1.30 + 1.30i)3-s + (−0.707 − 0.707i)4-s + (−0.707 − 1.70i)6-s + (0.923 − 0.382i)8-s − 2.41i·9-s + 1.41i·11-s + 1.84·12-s + (1.30 + 1.30i)13-s + i·16-s + (2.23 + 0.923i)18-s + 19-s + (−1.30 − 0.541i)22-s + (−0.707 + 1.70i)24-s + (−1.70 + 0.707i)26-s + (1.84 + 1.84i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5428374929\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5428374929\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 - 0.923i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
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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
−9.598504483326141121893887298845, −9.378385113773189503585689094019, −8.402915456919060621523404008213, −7.15986461244323053876872965652, −6.58491105492766475116640488527, −5.84426513795231623111247926015, −5.02270155842173468645848244939, −4.42103696348227727680075680467, −3.70952165612971071823472358277, −1.41162766731036922469977169289,
0.65401918380539009953019028975, 1.40899582786016670571648708225, 2.80330009239656723042943288687, 3.73220990931999586839118778391, 5.27866758275300040494709236093, 5.65749219634751646908656091491, 6.57523839562193699588624570310, 7.57846433849887001016489910264, 8.170655604688895261948583111744, 8.820495533903560994052642112998