L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s − 0.585i·11-s + (−4 − 4i)13-s − 1.00·14-s − 1.00·16-s + (0.585 + 0.585i)17-s − 2.82i·19-s + (−0.414 + 0.414i)22-s + (4.82 − 4.82i)23-s + 5.65i·26-s + (0.707 + 0.707i)28-s − 0.828·29-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s − 0.176i·11-s + (−1.10 − 1.10i)13-s − 0.267·14-s − 0.250·16-s + (0.142 + 0.142i)17-s − 0.648i·19-s + (−0.0883 + 0.0883i)22-s + (1.00 − 1.00i)23-s + 1.10i·26-s + (0.133 + 0.133i)28-s − 0.153·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5667641680\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5667641680\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + 0.585iT - 11T^{2} \) |
| 13 | \( 1 + (4 + 4i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.585 - 0.585i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 + (-4.82 + 4.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + (6.24 - 6.24i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.17iT - 41T^{2} \) |
| 43 | \( 1 + (6.07 + 6.07i)T + 43iT^{2} \) |
| 47 | \( 1 + (-9.24 - 9.24i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.58 - 2.58i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 + (8.41 - 8.41i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.17iT - 71T^{2} \) |
| 73 | \( 1 + (7.07 + 7.07i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.65iT - 79T^{2} \) |
| 83 | \( 1 + (0.828 - 0.828i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + (-4.58 + 4.58i)T - 97iT^{2} \) |
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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
−8.425770819406378904364415093875, −7.58222819550727480380726735745, −7.08939389483204367684743247361, −6.07452319785739702541658928777, −5.01938473869581091185097159526, −4.47258105463508237614342821531, −3.17122841109996786359978097928, −2.66455459517936266052932394310, −1.36138840676851781010874336715, −0.21224958927974796384427538446,
1.46005722063724427455159793663, 2.33687392539710334173977200568, 3.58106677386434096493193452698, 4.66484310720473843057625193760, 5.27652924173928646324344956071, 6.11388273049151311751016322272, 7.11879010609637318839229481865, 7.37585080680054045077033100163, 8.335887891779964536809139244289, 9.115188106968736645452974274440