L(s) = 1 | + 4i·2-s + 2·3-s − 8·4-s + 17i·5-s + 8i·6-s − 20i·7-s − 23·9-s − 68·10-s + 32i·11-s − 16·12-s + 80·14-s + 34i·15-s − 64·16-s + 13·17-s − 92i·18-s + 30i·19-s + ⋯ |
L(s) = 1 | + 1.41i·2-s + 0.384·3-s − 4-s + 1.52i·5-s + 0.544i·6-s − 1.07i·7-s − 0.851·9-s − 2.15·10-s + 0.877i·11-s − 0.384·12-s + 1.52·14-s + 0.585i·15-s − 16-s + 0.185·17-s − 1.20i·18-s + 0.362i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.396451 - 1.30939i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.396451 - 1.30939i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 - 4iT - 8T^{2} \) |
| 3 | \( 1 - 2T + 27T^{2} \) |
| 5 | \( 1 - 17iT - 125T^{2} \) |
| 7 | \( 1 + 20iT - 343T^{2} \) |
| 11 | \( 1 - 32iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 13T + 4.91e3T^{2} \) |
| 19 | \( 1 - 30iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 78T + 1.21e4T^{2} \) |
| 29 | \( 1 - 197T + 2.43e4T^{2} \) |
| 31 | \( 1 + 74iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 227iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 165iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 156T + 7.95e4T^{2} \) |
| 47 | \( 1 - 162iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 93T + 1.48e5T^{2} \) |
| 59 | \( 1 - 864iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 145T + 2.26e5T^{2} \) |
| 67 | \( 1 - 862iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 654iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 215iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 76T + 4.93e5T^{2} \) |
| 83 | \( 1 - 628iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 266iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 238iT - 9.12e5T^{2} \) |
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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
−13.59607737309315724784886601890, −11.83007192361508393306535944457, −10.72420058292812594306114924541, −9.862713283273342184544010865529, −8.363613101597092188964043044616, −7.45620588077621232973791973388, −6.83000658517553618462641463910, −5.82263150630822782155823716768, −4.16642901205392821823134307539, −2.64845798473429096116732732845,
0.58393472699915772495894284526, 2.13530020034696233179555476573, 3.34975771531299576180588028786, 4.83797653620749561727780672107, 5.96285018364300971255046241847, 8.276401013218958246843191233252, 8.845837243218007598649097974825, 9.579865359672247773109724927646, 10.97624425685648443780676057681, 11.92537294894514125911929604550