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 + i·9-s − 1.00i·14-s − 1.00·16-s + (0.707 + 0.707i)18-s + (−1.41 − 1.41i)23-s + (−0.707 − 0.707i)28-s + 2i·29-s + (−0.707 + 0.707i)32-s + 1.00·36-s + (1.41 + 1.41i)43-s − 2.00·46-s + ⋯ |
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 + i·9-s − 1.00i·14-s − 1.00·16-s + (0.707 + 0.707i)18-s + (−1.41 − 1.41i)23-s + (−0.707 − 0.707i)28-s + 2i·29-s + (−0.707 + 0.707i)32-s + 1.00·36-s + (1.41 + 1.41i)43-s − 2.00·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.369092322\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.369092322\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.65776646749660624276229901750, −10.04061072221709942574238249893, −8.810051274616026849948882966244, −7.86621014943965012687747934930, −6.87180082471471411966888682280, −5.73364110536209345822163853664, −4.76388419558164005128833273915, −4.11917056743580946188367879326, −2.72053426732590762748032989463, −1.54486986127973143592217368990,
2.21516036612903868742917252508, 3.58245483247344544841672958947, 4.45522278145457614265064462552, 5.70007574692904338507772889874, 6.10386607404840605350932034513, 7.36591469182560150197083690031, 8.065485621490381337846663777782, 8.973922913065695820493529002068, 9.745418629209944116986820700641, 11.16783967071500255630090898589