L(s) = 1 | + 1.41·3-s + 1.41i·5-s + (2.44 + i)7-s − 0.999·9-s + 3.46i·11-s − 4.24i·13-s + 2.00i·15-s + 4.89i·17-s + 4.24·19-s + (3.46 + 1.41i)21-s + 2.99·25-s − 5.65·27-s − 4.89·31-s + 4.89i·33-s + (−1.41 + 3.46i)35-s + ⋯ |
L(s) = 1 | + 0.816·3-s + 0.632i·5-s + (0.925 + 0.377i)7-s − 0.333·9-s + 1.04i·11-s − 1.17i·13-s + 0.516i·15-s + 1.18i·17-s + 0.973·19-s + (0.755 + 0.308i)21-s + 0.599·25-s − 1.08·27-s − 0.879·31-s + 0.852i·33-s + (−0.239 + 0.585i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.390390847\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.390390847\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.44 - i)T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 9.79iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 - 4.24iT - 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−9.406275519473745074543574834457, −8.488370955572869115235454490486, −7.88543950598168143767426346993, −7.38379262890616722366687162927, −6.18259445570200633448987071388, −5.38881414204191861355172523896, −4.43388756082497860331265551364, −3.28417549308225109353295217582, −2.60221142640196435238869658943, −1.53899649131809756130675819413,
0.839749707816597579851495690562, 2.07598010670868475954914178983, 3.14961474836494365467544485114, 4.08819261718751850089705381301, 5.02364604703206545426476663151, 5.72042792777604312732900515247, 7.00672826973844984424279272932, 7.68001035929391128784041090988, 8.482529236005793594293005084827, 9.034441201680729606504965925759