L(s) = 1 | + (0.541 + 0.541i)2-s − 0.414i·4-s + (0.541 − 0.541i)7-s + (0.765 − 0.765i)8-s + i·11-s + (1.30 + 1.30i)13-s + 0.585·14-s + 0.414·16-s + (−1.30 − 1.30i)17-s + (−0.541 + 0.541i)22-s + 1.41i·26-s + (−0.224 − 0.224i)28-s + 1.41·31-s + (−0.541 − 0.541i)32-s − 1.41i·34-s + ⋯ |
L(s) = 1 | + (0.541 + 0.541i)2-s − 0.414i·4-s + (0.541 − 0.541i)7-s + (0.765 − 0.765i)8-s + i·11-s + (1.30 + 1.30i)13-s + 0.585·14-s + 0.414·16-s + (−1.30 − 1.30i)17-s + (−0.541 + 0.541i)22-s + 1.41i·26-s + (−0.224 − 0.224i)28-s + 1.41·31-s + (−0.541 − 0.541i)32-s − 1.41i·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.810256925\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.810256925\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - iT \) |
good | 2 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 7 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 13 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 17 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.096921749779206244115097222525, −8.339495203300425073745399861156, −7.13513695893704173562104069609, −6.90505600475262116384819323756, −6.13440697728805897685587459421, −5.01307430904010468044314005182, −4.50407705428702326371130191259, −3.88080472047048408575633903594, −2.25971398872415830598843517695, −1.26724839068340565939684362076,
1.44565075872771602103422566681, 2.59481377297464390232789118007, 3.38842061553483943970491883859, 4.14042199117016341170988219727, 5.08271094781082793013481544356, 5.90529141545165679508387683184, 6.58680040849886704349564709976, 7.965203328190634315474074884203, 8.407344104181039009690419329063, 8.681570301686896495575043642143