L(s) = 1 | + (−0.923 − 0.382i)2-s + (1.30 + 1.30i)3-s + (0.707 + 0.707i)4-s + (−0.707 − 1.70i)6-s + (−0.382 − 0.923i)8-s + 2.41i·9-s + 1.41i·11-s + 1.84i·12-s + (1.30 − 1.30i)13-s + i·16-s + (0.923 − 2.23i)18-s − 19-s + (0.541 − 1.30i)22-s + (0.707 − 1.70i)24-s + (−1.70 + 0.707i)26-s + (−1.84 + 1.84i)27-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (1.30 + 1.30i)3-s + (0.707 + 0.707i)4-s + (−0.707 − 1.70i)6-s + (−0.382 − 0.923i)8-s + 2.41i·9-s + 1.41i·11-s + 1.84i·12-s + (1.30 − 1.30i)13-s + i·16-s + (0.923 − 2.23i)18-s − 19-s + (0.541 − 1.30i)22-s + (0.707 − 1.70i)24-s + (−1.70 + 0.707i)26-s + (−1.84 + 1.84i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.202410270\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.202410270\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.923 + 0.382i)T \) |
| 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
−9.549290647438705855246011974183, −8.856693998678244023882765798082, −8.271814660746980113071034843154, −7.72160249400714515255961300478, −6.70204724527445656439742935505, −5.38059162060606303561619986827, −4.23783040191009960272644619732, −3.62876461684270382202729981966, −2.72472027568448686381534945357, −1.83049524918377246948289925169,
1.11433052617015332319862087922, 1.96924812133161843190386158484, 2.98553796850816700673536892321, 3.95121348399835033493902104765, 5.75527321953177669088138829452, 6.56325545125810467420952774761, 6.82512687009536763772452383368, 8.024690108818522441651771848254, 8.381279074766846867721174095129, 8.908090324624514864140297186093